/r tA VECTOR CALCULUS WITH APPLICATIONS TO PHYSICS BY JAMES BYRNIE SHAW PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ILLINOIS ILLUSTRATED NEW YORK D. VAN NOSTRAND COMPANY Eight Warren Street 1922 s Copyright, 1922 By D. Van Nostrand Company All rights reserved, including that of translation into foreign languages, including the Scandinavian Printed in the United States of America PREFACE. This volume embodies the lectures given on the subject to graduate students over a period of four repetitions. The point of view is the result of many years of consideration of the whole field. The author has examined the various methods that go under the name of Vector, and finds that for all purposes of the physicist and for most of those of the geometer, the use of quaternions is by far the simplest in theory and in practice. The various points of view are mentioned in the introduction, and it is hoped that the es- sential differences are brought out. The tables of com- parative notation scattered through the text will assist in following the other methods. The place of vector work according to the author is in the general field of associative algebra, and every method so far proposed can be easily shown to be an imperfect form of associative algebra. From this standpoint the various discussions as to the fundamental principles may be under- stood. As far as the mere notations go, there is not much difference save in the actual characters employed. These have assumed a somewhat national character. It is un- fortunate that so many exist. The attempt in this book has been to give a text to the mathematical student on the one hand, in which every physical term beyond mere elementary teims is carefully defined. On the other hand for the physical student there will be found a large collection of examples and exercises which will show him the utility of the mathematical meth- ods. So very little exists in the numerous treatments of the day that does this, and so much that is labeled vector iii 505384 IV PREFACE analysis is merely a kind of short-hand, that it has seemed very desirable to show clearly the actual use of vectors as vectors. It will be rarely the case in the text that any use of the components of vectors will be found. The triplexes in other texts are very seldom much different fiom the ordi- nary Cartesian forms, and not worth learning as methods. The difficulty the author has found with other texts is that after a few very elementary notions, the mathematical student (and we may add the physical student) is suddenly plunged into the profundities of mathematical physics, as if he were familiar with them. This is rarely the case, and the object of this text is to make him familiar with them by easy gradations. It is not to be expected that the book will be free from errors, and the author will esteem it a favor to have all errors and oversights brought to his attention. He desires to thank specially Dr. C. F. Green, of the University of Illinois, for his careful assistance in reading the proof, and for other useful suggestions. Finally he has gathered his material widely, and is in debt to many authors for it, to all of whom he presents his thanks. James Byrnie Shaw. Urbana, III., July, 1922. TABLE OF CONTENTS. Chapter I. Introduction 1 Chapter II. Scalar Fields 18 Chapter III. Vector Fields 23 Chapter IV. Addition of Vectors 52 Chapter V. Vectors in a Plane 62 Chapter VI. Vectors in Space 94 Chapter VII. Applications 127 1. The Scalar of two Vectois 127 2. The Vector of two Vectors 136 3. The Scalar of three Vectors 142 4. The Vector of three Vectors 143 Chapter VIII. Differentials and Integrals 145 1. Differentiation as to one Scalar Parameter .... 145 Two Parameters 151 2. Differentiation as to a Vector 155 3. Integration 196 Chapter IX. The Linear Vector Function 218 Chapter X. Deformable Bodies 253 Strain 253 Kinematics of Displacement 265 Stress 269 Chapter XL Hydrodynamics 287 VECTOR CALCULUS CHAPTER I INTRODUCTION 1. Vector Calculus. By this term is meant a system of mathematical thinking which makes use of a special class of symbols and their combinations according to certain given laws, to study the mathematical conclusions resulting from data which depend upon geometric entities called vectors, or physical entities representable by vectors, or more generally entities of any kind which could be repre- sented for the purposes under discussion by vectors. These vectors may be in space of two or three or even four or more dimensions. A geometric vector is a directed segment of a straight line. It has length (including zero) and direc- tion. This is equivalent to saying that it cannot be de- fined merely by one single numerical value. Any problem of mathematics dependent upon several variables becomes properly a problem in vector calculus. For instance, analytical geometry is a crude kind of vector calculus. Several systems of vector calculus have been devised, differing in their fundamental notions, their notation, and their laws of combining the symbols. The lack of a uniform. notation is deplorable, but there seems little hope of the adoption of any uniform system soon. Existing systems have been rather ardently promoted by mathematicians of the same nationality as their authors, and disagreement exists as to their relative simplicity, their relative directness, and their relative logical exactness. These disagreements arise sometimes merely with regard to the proper manner of representing certain combinations of the symbols, or other matters which are purely matters of convention; 1 2 YKCTOR CALCULUS sometimes they are due to different views as to what are the import an1 things to find expressions for; and sometimes they are due to more fundamental divergences of opinion as to the real character of the mathematical ideas underlying any system of this sort. We will in- dicate these differences and dispose of them in this work. 2. Bases. We may classify broadly the various systems of vector calculus as geometric and algebraic. The former is to be found wherever the desire is to lay emphasis on the spatial character of the entities we are discussing, such as the line, the point, portions of a plane, etc. The latter lays emphasis on the purely algebraic character of the entities with which the calculations are made, these entities being similar to the positive and negative, and the imag- inary of ordinary algebra. For the geometric vector systems, the symbolism of the calculus is really nothing more than a short-hand to enable one to follow certain operations upon real geometric elements, with the possi- bility kept always in mind that these entities and the operations may at any moment be called to the front to take the place of their short-hand representatives. For the algebraic systems, the symbolism has to do with hypernumbers, that is, extensions of the algebraic negative and imaginary numbers, and does not pretend to be the translation of actual operations which can be made visible, any more than an ordinary calculation of algebra could be paralleled by actual geometric or physical operations. If these distinctions are kept in mind the different points of view become intelligible. The best examples of geo- metric systems are the Science of Extension of Grassmann, with its various later forms, the Geometry of Bynames of Study, the Geometry of Lines of Saussure, and the Geometry of Feuillets of Cailler. The best examples of algebraic systems are the Quaternions of Hamilton, Dyadics of Gibbs, INTRODUCTION ,3 Multenions of McAulay, Biquaternions of Clifford, Tri- quaternions of Combebiac, Linear Associative Algebra of Peirce. Various modifications of these exist, and some mixed systems may be found, which will be noted in the proper places. The idea of using a calculus of symbols for writing out geometric theorems perhaps originated with Leibniz, 1 though what he had in mind had nothing to do with vector calculus in its modern sense. The first effective algebraic vector calculus was the Quaternions of Hamilton 2 (1843), the first effective geometric vector calculus was the Ausdehn- ungslehre of Grassmann 3 (1844). They had predecessors worthy of mention and some of these will be noticed. 3. Hypernumbers. The real beginning of Vector Cal- culus was the early attempt to extend the idea of number. The original theory of irrational number was metric, 4 and defined irrationals by means of the segments of straight lines. When to this was added the idea of direction, so that the segments became directed segments, what we now call vectors, the numbers defined were not only capable of being irrational, but they also possessed quality, and could be negative or positive. Ordinary algebra is thus the first vector calculus. If we consider segments with direction in a plane or in space of three dimensions, then we may call the numbers they define hypernumbers. The source of the idea was the attempt to interpret the imaginary which had been created to furnish solutions for any quadratic or cubic. The imaginary appears early in Cardan's work. 5 For instance he gives as solution of the problem of separating 10 into two parts whose product is 40, the values 5 + V — 15, and 5 — V — 15. He considered these numbers as impossible and of no use. Later it was dis- covered that in the solution of the cubic by Cardan's formula there appeared the sum of two of these impossible 4 VWCTOfl CALCULUS values when the answer actually was real. Bombelli #;ive as the solution of the cubic r 3 = 15x + 4 the form ^(2 + V - 121) + ^(2 - V - 121) = 4. These impossible numbers incited much thought and there came about several attempts to account for them and to interpret them. The underlying question was essen- tially that of existence, which at that time was usually sought for in concrete cases. The real objection to the negative number was its inapplicability to objects. Its use in a debit and credit account would in this sense give it existence. Likewise the imaginary and the complex num- ber, and later others, needed interpretation, that is, applica- tion to physical entities. 4. Wessel, a Danish surveyor, in 1797, produced a satisfactory method 7 of defining complex numbers by means of vectors in a plane. This same method was later given by Argand 8 and afterwards by Gauss 9 in connection with various applications. Wessel undertook to go farther and in an analogous manner define hypernumbers by means of directed segments, or vectors, in space of three dimen- sions. He narrowly missed the invention of quaternions. In 1813 Servois 10 raised the question whether such vectors might not define hypernumbers of the form . p cos a + q cos (3 + r cos y and inquired what kind of non-reals p, q, r would be. He did not answer the question, however, and Wessel's paper remained unnoticed for a century. 5. Hamilton gave the answer to the question of Servois as the result of a long investigation of the whole problem. 11 He first considered algebraic couples, that is to say in our terminology, hypernumbers needing two ordinary numerical INTRODUCTION 5 values to define them, and all possible modes of combining them under certain conditions, so as to arrive at a similar couple or hypernumber for the product. He then con- sidered triples and sets of numbers in general. Since — 1 and i = V — 1 are roots of unity, he paid most attention to definitions that would lead to new roots of unity. His fundamental idea is that the couple of numbers (a, b) where a and b are any positive or negative numbers, rational or irrational, is an entity in itself and is therefore subject to laws of combination just as are single numbers. For instance, we may combine it with the other couple (x, y) in two different ways : (a, b) + (x, y) = (a + x, b + y) (a, 6) X (x, y) = {ax — by, ay + bx). In the first case we say we have, added the couples, in the second case that we have multiplied them. It is possible to define division also. In both cases if we set the couple on the right hand side equal to {u, v) we find that dujdx — dv/dy, dujdy = — dv/dx. Pairs of functions u, v which satisfy these partial differential equations Hamilton called conjugate functions. The partial differential equations were first given by Cauchy in this connection. The particular couples €l = (1, 0), € 2 = (0, 1) play a special role in the development, for, in the first place, any couple may be written in the form (a, b) = aei + be 2 and the notation of couples becomes superfluous; in the second place, by defining the products of ei and e 2 in various ways we arrive at various algebras of couples. The general C> VECTOR CALCULUS definition would be, using the • for X, €l'€i = Cin€i + Cii 2 € 2 , €i'€ 2 = Ci2i€i + ^12262, €2'€i = C2ll€i + C212€2> «2 * €2 = C221«l + C222€2- By varying the choice of the arbitrary constants c, and Hamilton considered several different cases, different algebras of couples could be produced. In the case above the c's are all zero except Cm = 1, C122 = 1, C212 — 1, C221 = — 1. From the character of 4 it may be regarded as entirely identical with ordinary 1, and it follows therefore that e 2 may be regarded as identical with the V — 1. On the other hand we may consider €1 to be a unit vector pointing to the right in the plane of vectors, and c 2 to be a unit vector perpendicular to ei. We have then a vector calculus practically identical with Wessel's. The great merit of Hamilton's investigation lies of course in its generality. He continued the study of couples by a similar study of triples and then quadruples, arriving thus at Quaternions. His chief difference in point of view from those who followed him and who used the concept of couple, triple, etc. {Mul- tiple we will say for the general case), is that he invariably defined one product, whereas others define usually several. 6. Multiples. There is a considerable tendency in the current literature of vector calculus to use the notion of multiple. A vector is usually designated by a triple as (x, y, z), and usually such triple is called a vector. It is generally tacitly understood that the dimensions of the numbers of the triple are the same, and in fact most of the products defined would have no meaning unless this homogeneity of dimension were assumed to hold. We find products defined arbitrarily in several ways. For instance, the scalar product of the triples (a, b, c) and (x, y, z) INTRODUCTION 7 is =fc (ax + by + cz), the sign depending upon the person giving the definition; the vector product of the same two triples is usually given as the triple (bz — cy, ex — az, ay — bx). It is obvious at once that a great defect of such definitions is that the triples involved have no sense until the significance of the first number, the second number, and the third number in each triple is understood. If these depend upon axes for their meaning, then the whole calculus is tied down to such axes, unless, as is usually done, the expressions used in the definitions are so chosen as to be in some respects independent of the particular set of axes chosen. When these expressions are thus chosen as invariants under given transformations of the axes we arrive at certain of the well-known systems of vector analysis. The transformations usually selected to furnish the profitable expressions are the group of orthog- onal transformations. For instance, it was shown by Burkhardt 12 that all the invariant expressions or invariant triples are combinations of the three following : ax + by + cz, (bz — cy, ex — az, ay — bx), (al + bm + cn)x + (am — bl)y + (an — cl)z, (bl — am)x + (al -f- bm + cn)y + (bn — cm)z, (cl — ari)x + (cm — bn)u + (al + bm + cn)z. A study of vector systems from this point of view has been made by Schouten. 13 7. Quaternions. In his first investigations, Hamilton was chiefly concerned with the creation of systems of hypernumbers such that each of the defining units, similar to the ei and € 2 above, was a root of unity. 14 That is, the process of multiplication by iteration would bring back the multiplicand. He was actually interested in certain special 8 VECTOR CALCULUS cases of abstract groups, 15 and if he had noticed the group property his researches would perhaps have extended into the whole field of abstract groups. In quaternions he found a set of square roots of — 1, which he designated by i, j, k, connected with his triples though belonging to a set of quad- ruples. In his Lectures on Quaternions, the first treatise he published on the subject, he chose a geometrical method of exposition, consequently many have been led to think of quaternions as having a geometric origin. However, the original memoirs show that they were reached in a purely algebraic way, and indeed according to Hamilton's philoso- phy were based on steps of time as opposed to geometric steps or vectors. The geometric definition is quite simple, however, and not so abstract as the purely algebraic definition. Ac- cording to this idea, numbers have a metric definition, a number, or hypernumber, being the ratio of two vectors. If the vectors have the same direction we arrive at the ordinary numerical scale. If they are opposite we arrive at the negative numbers. If neither in the same direction nor opposite we have a more general kind of number, a hypernumber in fact, which is a quaternion, and of which the ordinary numbers and the negative numbers are merely special cases. If we agree to consider all vectors which are parallel and in the same direction as equivalent, that is, call them free vectors, then for every pair of vectors from the origin or any fixed point, there is a quaternion. Among these quaternions relations will exist, which will be one of the objects of study of later chapters. 8. Mobius was one of the early inventors of a vector calculus on the geometric basis. In his Barycentrisch.es Kalkul 16 he introduced a method of deriving points from other points by a process called addition, and several INTRODUCTION 9 applications were made to geometry. The barycentric calculus is somewhat between a system of homogeneous coordinates and a real vector calculus. His addition was used by Grassmann. 9. Grassmann in 1844 published his treatise called Die lineale Ausdehnungslehre 17 in which several different proc- esses called multiplication are used for the derivation of geometric entities from other geometric entities. These processes make use of a notation which is practically a sort of short-hand for the geometric processes involved. Grassmann considered these various kinds of multiplication abstractly, leaving out of account the meaning of the elements multiplied. His methods apply to space of N dimensions. In the symmetric multiplication it is possible to interchange any two of the factors without affecting the result. In the circular multiplication the order may be changed cyclically. In the lineal multiplication all the laws hold as well for any factors which are linear combina- tions of the hypernumbers which define the base, as for those called the base. He studies two species of circular multiplication. If the defining units of the base are ex, e 2 , e 3 • • •€„, then we have in the first variety of circular multipli- cation the laws €l 2 + € 2 2 + 6 3 2 + • • • + € n 2 = 0, €i€j = €j€i. In the second variety we have the laws ei 2 = 0, e/ = 0, • - • e n 2 = 0, Mi = 0, *+j. In the lineal genus of multiplication he studies two species, in the first, called the algebraic multiplication, we have the law My = *fii for all i, j. while in the second, called the exterior multiplication, the interchange of any two factors changes the sign of the 10 VECTOR CALCULUS result. Of the latter there are two varieties, the progressive multiplication in which the number of dimensions of the geometric figure which is the product is the sum of the dimensions of the factors, while in the other, called re- gressive multiplication, the dimension of the product is the difference between the sum of the dimensions of the factors and N the dimension of the space in which the operation takes place. From the two varieties he deduces another kind called interior multiplication. If we confine our thoughts to space of three dimensions, defined by points, and if €1, e 2 , e 3 , e 4 are such points, the progressive exterior product of two, as €1, e 2 , is ei€ 2 and represents the segment joining them if they do not coincide. The product is zero if they coincide. The product of this into a third point € 3 is ei€ 2 e 3 and represents the parallelogram with edges €162, ei€ 3 and the other two parallel to these respectively. If all three points are in a straight line the product is zero. The exterior progressive product c 1 e 2 e 3 € 4 represents the parallelepiped with edges €ie 2 , €ie 3 , €i€ 4 and the opposite parallel edges. The regressive exterior product of €i€ 2 and €ie 3 € 4 is their common point €1. The regressive product of €ie 2 e 3 and €ie 2 € 4 is their common line €ie 2 . The complement of €1 is defined to be € 2 e 3 e 4 , and of €i€ 2 is e 3 fct, and of €i€ 2 e 3 is € 4 . The interior product of any expression and another is the progressive or regressive product of the first into the complement of the other. For instance, the interior product of €1 and e 2 is the progressive product of €1 and €i€ 3 e 4 which vanishes. The interior product of e 2 and e 2 is the product of e 2 and eie 3 e 4 which is € 2 eie 3 e 4 . The interior product of €j€ 2 e 3 and ei€ 4 is the product of €ie 2 e 3 and € 2 e 3 which would be regressive and be the line e 2 e 3 . We have the same kinds of multiplication if the expres- sions e are vectors and not points, and they may even be INTRODUCTION 1 1 planes. The interpretation is different, however. It is easy to see that Grassmann's ideas do not lend themselves readily to numerical application, as they are more closely related to the projective transformations of space. In fact, when translated, most of the expressions would be phrased in terms of intersections, points, lines and planes, rather than in terms of distances, angles, areas, etc. 10. Dyadics were invented by Gibbs, 18 and are of both the algebraic and the geometric character. Gibbs has, like Hamilton, but one kind of multiplication. If we have given two vectors a, (3 from the same point, their dyad is a(3. This is to be looked upon as a new entity of two dimensions belonging to the point from which the vectors are drawn. It is not a plane though it has two dimensions, but is really a particular and special kind of dyadic, an entity of two- dimensional character, such that in every case it can be considered to be the sum of not more than three dyads. Gibbs never laid any stress on the geometric existence of the dyadic, though he stated definitely that it was to be considered as a quantity. His greatest stress, however, was upon the operative character of the dyadic, its various combinations with vectors being easily interpretable. The simplest interpretation is from its use in physics to represent strain. Gibbs also pushed his vector calculus into space of many dimensions, and into triadic and higher forms, most of which can be used in the theory of the elasticity of crystals. The scalar and vector multiplication he considered as functions of the dyadic, rather than as multiplications, and there are corresponding functions of triadics and higher forms. In this respect his point of view is close to that of Hamilton, the difference being in the use of the dyadic or the quaternion. 11. Other forms of vector calculus can be reduced to 3 12 VECTOR CALCULUS these or to combinations of parts of these. The differences are usually in the notations, or in the basis of exposition. Notations for One Vector Greek letters, Hamilton, Tait, Joly, Gibbs. Italics, Grassmann,_Peano, Fehr, Ferraris, Macfarlane. Heun writes a, b, c. Old English or German letters, Maxwell, Jaumann, Jung, Foppl, Lorentz, Gans, Abraham, Bucherer, Fischer, Sommerfeld. Clarendon type, Heaviside, Gibbs, Wilson, Jahnke, Timer- ding, Burali-Forti, Marcolongo. Length of a vector T ( ), Hamilton, Tait, Joly. | | , Gans, Bucherer, Timerding. Italic corresponding to the ve ctor letter, Wilson, Jaumann, &ing, Fischer, Jahnke. Corresponding small italic, Macfarlane. Mod. ( ), Peano, Burali-Forti, Marcolongo, Fehr. Unit of a vector U ( ), Hamilton, Tait, Joly, Peano. Clarendon small, Wilson. ( )i, Bucherer, Fischer. Corresponding Greek letter, Macfarlane. Some write the vector over the length. Square of a vector ( ) 2 . The square is usually positive except in Quaternions, where it is negative. Reciprocal ( ) -1 , Hamilton, Tait, Joly, Jaumann. tt , Hamilton, Tait, Joly, Fischer, Bucherer. CHAPTER II SCALAR FIELDS 1. Fields. If we consider a given set of elements in space, we may have for each element one or more quantities determined, which can be properly called functions of the element. For instance, at each point in space we may have a temperature, or a pressure, or a density, as of the air. Or for every loop that we may draw in a given space we may have a length, or at some fixed point a potential due to the loop. Again, we may have at each point in space a velocity which has both direction and length, or an electric intensity, or a magnetic intensity. Not to multiply examples unnecessarily, we can see that for a given range of points, or lines, or other geometric elements, we may have a set of quantities, corresponding to the various elements of the range, and therefore constituting a function of the range, and these quantities may consist of numerical values, or of vectors, or of other hypernumbers. When they are of a simple numerical character they are called scalars, and the function resulting is a scalar function. Examples are the density of a fluid at each point, the density of a distribution of energy, and similar quantities consisting of an amount of some entity per cubic centimeter, or per square centimeter, or per centimeter. EXAMPLES (1) Electricity. The unit of electricity is the coulomb, connected with the absolute units by the equations 1 coulomb = 3 • 10° electrostatic units == 10 -1 electromagnetic units. 13 14 VECTOR CALCULUS The density of electricity is its amount in a given volume, area, op length divided by the volume, area, or length respectively. The dimensions of electricity will be repre- sented by [9], and for its amount the symbol 9 will be used. For the volume density we will use e, for areal density e' , for linear density e". If the distribution may be considered to be continuous, we may take the limits and find the density at a point. (2) Magnetism. Considering magnetism to be a quan- tity, we will use for the unit of measurement the maxwell, connected with the absolute units by the equation 1 maxwell = 3-10 10 electrostatic units = 1 electromagnetic unit. Sometimes 10 8 maxwells is called a weber. The symbol for magnetism will be $, the dimensions [$], the densities m, m', m". (3) Action. This quantity is much used in physics, the principle of least action being one of the most important fundamental bases of modern physics. The dimensions of action are [93>], the symbol we shall use is A, and the unit might be a quantum, but for practical purposes a joule-second is used. In the case of a moving particle the action at any point depends upon the path by which the particle has reached the point, so that as a function of the points of space it has at each point an infinity of values. A function which has but a single value at a point will be called monodromic, but if it has more than one value it will be called polydromic. The action is therefore a polydromic function. We not only have action in the motion of par- ticles but we find it as a necessary function of a momentum field, or of an electromagnetic field. (4) Energy. The unit of energy is the erg or the joule SCALAR FIELDS 15 = 10 7 ergs. Its dimensions are [G^T 7-1 ], its symbol will beW. (5) Activity. This should not be confused with action. It is measured in watts, symbol J, dimensions [Q$T~ 2 ]. (6) Energy-density. The symbol will be U, dimensions (7) Activity-density. The symbol will be Q, dimensions pi- 3 r 2 ]. (8) Mass. The symbol is M, dimensions [0$77r 2 ]. The unit of mass is the gram. A distribution of mass is usually called a distribution of matter. (9) Density of mass. The symbol will be c, dimensions (10) Potential of electricity. Symbol V, dimensions (11) Potential of magnetism. Symbol N, dimensions [02 7 - 1 ]. (12) Potential of gravity. Symbol P, dimensions [G^T 7-1 ]. 2. Levels. Points at which the function has the same value, are said to define a level surface of the function. It may have one or more sheets. Such surfaces are usually named by the use of the prefixes iso and equi. For instance, the surfaces in a cloud, which have all points at the same temperature, are called isothermal surfaces; surfaces which have points at the same pressure are called isobaric surfaces; surfaces of equal density are isopycnic surfaces; those of equal specific volume (reciprocal of the density) are the iso- steric surfaces; those of equal humidity are isohydric surfaces. Likewise for gravity, electricity, and magnetism we have equipotential surfaces. 3. Lamellae. Surfaces are frequently considered for which we have unit difference between the values of the function for the successive surfaces. These surfaces and 16 VECTOR CALCULUS the space between them constitute a succession of unit lamellae. If we follow a line from a point A to a point B, the number of unit lamellae traversed will give the difference between the two values of the function at the points A and B. If this is divided by the length of the path we shall have the mean rate of change of the function along the path. If the path is straight and the unit determining the lamellae is made to decrease indefinitely, the limit of this quotient at any point is called the derivative of the function at that point in the given direction. The derivative is ap- proximately the number of unit lamellae traversed in a unit distance, if they are close together. 4. Geometric Properties. Monodromic levels cannot in- tersect each other, though any one may intersect itself. Any one or all of the levels may have nodal lines, conical points, pinch-points, and the other peculiarities of geo- metric surfaces. These singularities usually depend upon the singularities of the congruence of normals to the surface. In the case of functions of two variables, the scalar levels will be curves on the surface over which the two variables are defined. Their singularities may be any that can occur in curves on surfaces. 5. Gradient. The equation of a level surface is found by setting the function equal to a constant. If, for in- stance, the point is located by the coordinates x, y, z and the function is f(x, y, z), then the equation of any level is u = /(*> V> z ) = C. If we pass to a neighboring point on the same surface we have du = f{x + dx, y -f- dy, z + dz) — f{x, y, z) = 0. We may usually find functions df/dx, bf\a\ df/dz, SCALAR FIELDS 17 functions independent of dx, dy, dz, such that du — dfjdx • dx + df/dy • dy + df/dz • dz. Now the vector from the first point to the second has as the lengths of its projections on the axes: dx, dy, dz; and if we define a vector whose projections are dfjdx, df/dy, df/dz, which we will call the Gradient of f, then the con- dition du = is the condition that the gradient of / shall be perpendicular to the differential on the surface. Hence, if we represent the gradient of / by v/, and the differential change from one point to the other by dp, we see that dp is any infinitesimal tangent on the surface and v/ is along the normal to the surface. It is easy to see that if we differen- tiate u in a direction not tangent to a level surface of u we shall have du = df/dx-dx + df/dy •<&,+ df/dz -dz = dC. If the length of the differential path is ds then we shall have* du/ds = projection of^fon the unit vector in the direction of dp. The length of the vector v/ is sometimes called the gradient rather than the vector itself. Sometimes the negative of the expression used here is called the gradient. When the three partial derivatives of / vanish for the same point, the intensity of the gradient, measured by its length, is zero, and the direction becomes indeterminate from the first differentials. At such points there are singu- larities of the function. At points where the function becomes infinite, the gradient becomes indeterminate and such points are also singular points. 6. Potentials. The three components of a vector at a point may be the three partial derivatives of the same function as to the coordinates, in which case the vector may be looked upon as the gradient of the integral func- * Since dxjds, dyjds, dzjds are the direction-cosines of dp. 18 VECTOR CALCULUS tion, which is called a potential junction, or sometimes a force function. For instance, if the components of the velocity satisfy the proper conditions, the velocity is the gradient of a velocity 'potential. These conditions will be discussed later, and the vector will be freed from dependence upon any axes. 7. Relative Derivatives. In case there are two scalar functions at a point, we may have use for the concept of the derivative of one with respect to the other. This is defined to be the quotient of the intensity of the gradient of the first by that of the second, multiplied by the cosine of their included angle. If the unit lamellae are constructed, it is easy to see from the definition that the relative deriva- tive of the first as to the second will be the limit of the average or mean of the number of unit sheets of the first traversed from one point to another, along the normal of the second divided by the number of unit sheets of the second traversed at the same time. For instance, if we draw the isobars for a given region of the United States and the simultaneous isotherms, then in passing from a point A to a point B if we traverse 24 isobaric unit sheets and 10 isothermal unit sheets, the average is 2.4 isobars per isotherm. ^ 8. Unit-Tubes. If there are two scalar functions in the field, and the unit lamellae are drawn, the unit sheets will usually intersect so as to divide the space under considera- tion into tubes whose cross-section will be a curvilinear parallelogram. Since the area of such parallelogram is approximately dsids2 esc 0, where dsi is the distance from a unit sheet of the function u to the next unit sheet, and ds 2 the corresponding distance for the function v, while 6 is the angle between the surfaces; and since we have, Tyu being the intensity of the gradient SCALAR FIELDS 19 of u, and T^/v the intensity of the gradient of v, dsi - 1/TVu, ds 2 = 1/Tw the area of the parallelogram will be l/(TyuTvv sin 6). Consequently if we count the parallelograms in any plane Fig. 1. cross-section of the two sets of level surfaces, this number is an approximate value of the expression T^uT^Jv sin 6 X area parallelogram when summed over the plane cross-section. That is to say, the number of these tubes which stand perpendicular to the plane cross-section is the approximate integral of the expression T^uT^v sin 6 over the area of the cross-section. These tubes are called unit tubes for the same reason that the lamellae are called unit lamellae. In counting the tubes it must be noticed whether the successive surfaces crossed correspond to an increasing or to a decreasing value of u or of v. It is also clear that when sin 6 is everywhere the integral must be zero. In such case the three Jacobians d(u, v)/d(y, z), d(u, v)/d(z, x), d{u, v)/d(x, y) 20 VECTOR CALCULUS are each equal zero, and this is the^condition that u is a function of v. In case the plane of cross-section is the x, y plane, the first two expressions vanish anyhow, since u, v are functions of x, y only. It is clear if we take the levels of one of the functions, say u, as the upper and lower parts of the boundary of the cross-section, that in passing from one of the other sides of the boundary along each level of u the number of unit tubes we encounter from that side of the boundary to the opposite side is the excess of the value of v on the second side over that on the first side. If then we count the dif- ferent tubes in the successive lamellae of u between the two sides of the cross-section we shall have the total excess of those on the second side over those on the first side. That is to say, the number of unit tubes or the integral over the area bounded by level 1 and level 2 of u, and any other two lines which cross these two levels so as to produce a simple area between, is the excess of the sum between the two levels of the values of v on one side over the same sum between the two levels of u on the other side. These graphical solutions are used in Meteorology. This gives the excess of the integral J vdu along the second line between the two levels of u, over the same in- tegral along the first line. It represents the increase of this integral in a change of path from one line to the other. For instance if the integral is energy, the number of tubes is the amount of energy stored or released in the passage from one line to the other, as in a cyclone. The number of tubes for any closed path is the approximate integral I rdu around the path. , SCALAR FIELDS 21 EXERCISES. 1. If the density varies as the distance from a given axis, what are the isopycnic surfaces? 2. A rotating fluid mass is in equilibrium under the force of gravity, the hydrostatic pressure, and the centrifugal force. What are the levels? Show that the field of force is conservative. 3. The isobaric surfaces are parallel planes, and the isopycnic surfaces are parallel planes at an angle of 10° with the isobaric planes. What is the rate of change of pressure per unit rate of change of density along a line at 45° with the isobaric planes? 4. If the pressure can be stated as a function of the density, what conditions are necessary? Are they sufficient? What is the interpreta- tion with regard to the levels? 5. Three scalar functions have a functional relation if their Jacobian vanishes. What does this mean with regard to their respective levels? 6. If the isothermal surfaces are spheres with center at the earth's center, the temperature sheets for decrease of one degree being 166.66 feet apart, and if the isobaric levels are similar spheres, the pressure being given by log B = log B, - 0.0000177 (a - z ), where B is the pressure at z feet above the surface of the earth, what is the relative derivative of the temperature as to the pressure, and the pressure as to the temperature? 7. To find the maximum of u(x, y, z) we set du = 0. If there is also a condition to be fulfilled, v(x, y, z) = 0, then dv = also. These two equations in dx, dy, dz must be satisfied for all compatible values of dx, dy, dz, and we must therefore have du du du _ _ dy # dv dv_ dx' dy' dz' ~ dx' dy' dz } which is equivalent to the single vector equation Vw = wyv. What does this mean in terms of the levels : ; The unit tubes? If there is also another equation of condition l(x, y, z) =0 then also dt = and the Jacobian of the three functions u, v, t must equal zero. Interpret. 8. On the line of intersection of two levels of two different functions the values of both functions remain constant. If we differentiate a third function along the locus in question, the differential vanishing everywhere, what is the significance? 22 VECTOR CALCULUS 9. If a field of force has a potential, then a fluid, subject to the force and such that its pressure is a function of the density and the tempera- ture, will have the equipotential levels for isobaric levels also. The density will be the derivative of the pressure relative to the potential. Show therefore that equilibrium is not possible unless the isothermals are also the levels of force and of pressure. [p = p(c, T), and vp = cvv = PcVc + prvT. If then vc = 0, cvv = prVT.] 10. If the full lines below represent the profiles of isobaric sheets, and the dotted lines the profiles of isosteric sheets, count the unit tubes between the two verticals, and explain what the number means. If they were equipotentials of gravity and isopycnic surfaces, what would the number of unit tubes mean? Fig. 2. 11. If u = y — 12x 3 and v = y + x 2 + \x, find Vw and w and TvuTw -sin 6, and integrate the latter over the area between x = f x = 1, y = 0, y = 12. Draw the lines. 12. If u = ax + by + cz and v = x 2 -f- if + z 2 , find vw and vv and TyuTvvsm 6 and integrate the latter expression over the surface of a cylinder whose axis is in the direction of the z axis. Find the deriva- tive of each relative to the other. CHAPTER III VECTOR FIELDS 1. Hypercomplex Quantity. In the measurement of quantity the first and most natural invention of the mind was the ordinary system of integers. Following this came the invention of fractions, then of irrational numbers. With these the necessary list of numbers for mere measure- ment of similar quantities is closed, up to the present time. Whether it will be necessary to invent a further extension of number along this line remains for the future to show. In the attempt to solve equations involving ordinary numbers, it became necessary to invent negative numbers and imaginary numbers. These were known and used as fictitious numbers before it was noticed that quantities also are of a negative or an "imaginary" character. We find instances everywhere. In debit and credit, for ex- ample, we have quantity which may be looked upon as of two different kinds, like iron and time, but the most logical conception is to classify debits and credits together in the single class balance. One's balance is what he is worth when the debits and credits have been compared. If the preponderance is on the side of debit we consider the balance negative, if on the side of credit we consider the balance positive. Likewise, we may consider motion in each direc- tion of the compass as in a class by itself, never using any conception of measurement save the purely numerical one of comparing things which are exactly of the same kind together. But it is more logical, and certainly more general, to consider motions in all directions of the compass and of any distances as all belonging to a single class of quantity. 23 24 VECTOR CALCULUS In that case the comparison of the different motions leads us to the notion of complex numbers. When Wessel made his study of the vectors in a plane he was studying the hypernumbers we usually call "the complex field." The hypernumbers had been studied in themselves before, but were looked upon (rightly) as being creations of the mind and (in that sense correctly) as having no existence in what might be called the real world. However, their deduction from the vectors in a plane showed that they were present as relations of quantities which could be considered as alike. Again when Steinmetz made use of them in the study of the relations of alternating currents and electromotive forces, it became evident that the so-called power current and wattless current could be regarded as parts of a single complex current, and similarly for the electromotive forces. The laws of Ohm and Kirchoff could then be generalized so as to be true for the new complex quantities. In this brief history we find an example of the interaction of the develop- ments of mathematics. The inventions of mathematics find instances in natural phenomena, and in some cases furnish new conceptions by which natural phenomena can be regarded as containing elements that would ordinarily be completely overlooked. In space of three (or more) dimensions, the vectors issuing from a point in all directions and of all lengths furnish quantities which may be considered to be all of the same kind, on one basis of classification. Therefore, they will define certain ratios or relations which may be called hypernumbers. This is the class of hypernumbers we are particularly concerned with, though we shall occa- sionally notice others. Further, any kind of quantity which can be represented completely for certain purposes by vectors issuing from a point we will call vector quantity. VECTOR FIELDS 25 Such quantities, for instance, are motions, velocities, accelerations, at least in the Newtonian mechanics, forces, momenta, and many others. The object of VECTOR CAL- CULUS is to study these hypernumbers in relation to their corresponding quantities, and to derive an algebra capable of handling them. We do not consider a vector as a mere triplex of ordinary numbers. Indeed, we shall consider two vectors to be identical when they represent or can represent the same quantity, even though one is ex- pressed by a certain triplex, as ordinary Cartesian coordinates, and the other by another triplex, as polar coordinates. The numerical method of defining the vector will be considered as incidental. 2. Notation. We shall represent vectors for the most part by Greek small letters. Occasionally, however, as in Electricity, it will be more convenient to use the standard symbols, which are generally Gothic type. As indicated on page 12 there is a great variety of notation, and only one principle seems to be used by most writers, namely that of using heavy type for vectors, whatever the style of type. In case the vector is from the origin to the point (x, y, z) it may be indicated by Px, y, z> while for the same point given by polar coordinates r, <p, 6 we may use Pr, <p, 6) In case a vector is given by its components as X, Y, Z we will indicate it by ?x, y, z 3. Equivalence. All vectors which have the same direc- tion and same length will be considered to be equivalent. Such vectors are sometimes called free vectors. The term vector will be used throughout this book, however, with no other meaning. 2G VECTOR CALCULUS In case vectors are equivalent only when they lie on the same line, and have the same direction and length, they will be called glissants. A force applied to a rigid body must be considered to be a glissant, not a vector. In case vectors are equivalent only when they start at the same point and coincide, they will be called radials. The resultant moment of a system of glissants with respect to a point A is a radial from A. The equivalence of two vectors a = implies the existence of equalities infinite in number, for their projections on any other lines will then be equal. The infinite set of equalities, however, is reducible in an infinity of ways to three independent equalities. For instance, we may write either a x = ft., ay = fi y , a 2 = 13 z , or a r = B r , a <p = ^ lf> ,a lf! = /?„. The equivalence of two glissants implies sets of equalities reducible in every case to five independent equalities. The equivalence of two radials reduces to sets of six equalities. 4. Vector Fields. Closely allied to the notion of radial is that of vector field. A vector field is a system of vectors each associated with a point of space, or a point of a surface, or a point of a line or curve. The vector is a function of the position of the point which is itself usually given by a vector, as p. The vector function may be monodromic or polydromic. We will consider some of the usual vector fields. EXAMPLES (1) Radius Vector, p [L]. This will usually be indicated by p. In case it is a function of a single parameter, as t, the points defined will lie on a curve;* in case it is a function * We are discussing mainly ordinary functions, not the "pathologic type." VECTOR FIELDS 27 of two parameters, u, v, the points defined will lie on a surface. The term vector was first introduced by Hamilton in this sense. When we say that the field is p, we mean that at the point whose vector is p measured from the fixed origin, there is a field of velocity, or force, or other quantity, whose value at the point is p. (2) Velocity, a [XT 7-1 ]. Usually we will designate veloc- ity by c. In the case of a moving gas or cloud, each particle has at each point of its path a definite velocity, so that we can describe the entire configuration of the moving mass at any instant by stating what function a is of p, that is, for the point at the end of the radius vector p assign the velocity vector. The path of a moving particle will be called a trajectory. At each point of the path the velocity a is a tangent of the trajectory. If we lay off from a fixed point the vectors a which corre- spond to a given trajectory, their terminal points will lie on a locus called by Hamilton the hodograph of the trajectory. For instance, the hodographs of the orbits of the planets are circles, to a first approximation. If we multiply a by dt, which gives it the dimensions of length, namely an infinitesimal length along the tangent of the trajectory, the differential equation of the trajectory becomes dp = adt. The integral of this in terms of t gives the equation of the trajectory. (3) Acceleration. t[LT~ 2 ]. An acceleration field is simi- lar to a velocity field except in dimensions. The accelera- tion is the rate of change of the vector velocity at a point, consequently, if a point describes the hodograph of a trajec- tory so that its radius vector at a given time is the velocity in the trajectory at that time, the acceleration will be a 3 L\S VECTOR CALCULUS tangent to the hodograph, and its length will be the velocity of the moving point in the hodograph. We will use r to indicate acceleration. (4) Momentum Density. T [$QL~ 4 ]. This is a vector function of points in space and of some number which can be attached to the point, called density. In the case of a moving cloud, for instance, each point of the cloud will have a velocity and a density. The product of these two factors will be a vector whose direction is that of the velocity and whose length is the product of the length of the velocity vector and the density. However, momentum density may exist without matter and without motion. In electro- dynamic fields, such as could exist in the very simple case of a single point charge of electricity and a single magnet pole at a point, we also have at every point of space a momentum density vector. This may be ascribed to the hypothetical motion of a hypothetical ether, but the essen- tial feature is the existence of the field. If we calculate the integral of the projection of the momentum density on the tangent to a given curve from a point A to a point B, the value of the integral is the action of an infinitesimal volume, an action density, along that path from A to B. The integration over a given volume would give the total action for all the particles over their various paths. This would be a minimum for the paths actually described as compared with possible paths. Specific momentum is momentum density of a moving mass. (5) Momentum. Y [TOL -1 ]. The volume integral of momentum density or specific momentum is momentum. Action is the line-integral of momentum. (6) Force Density. F [^QL^T- 1 ]. If a field of momen- tum density is varying in time then at each point there is a vector which may be called force-density, the time derivative VECTOR FIELDS 29 of the momentum density. Such cases occur in fields due to moving electrons or in the action of a field of electric intensity upon electric density, or magnetic intensity on magnetic density. (7) Force. X [mL- 1 ? 7 - 1 ]. The unit of force has re- ceived a name, dyne. It is the volume integral of force density. The time integral of a field of force is momentum. In a stationary field of force the line integral of the field for a given path is the difference in energy between the points at the ends of the path, or what is commonly called work. In case the field is conservative the integral has the same value for all paths (which at least avoid certain singular points), and depends only on the end points, This takes place when the field is a gradient field of a force- function, or a potential function. If we project the force upon the velocity at each point where both fields exist, the time integral of the scalar quantity which is the product of the intensity of the force, the intensity of the velocity and the cosine of the angle between them, is the activity at the point. (8) Flux Density. 12 [UT~ 1 }. In the case of the flow of an entity through a surface the limiting value of the amount that flows normally across an infinitesimal area is a vector whose direction is that of the outward normal of the surface, and whose intensity is the limit. In the case of a flow not normal to the surface across which the flux is to be de- termined, we nevertheless define the flux density as above. The flux across any surface becomes then the surface integral of the projection of the flux density on the normal of the surface across which the flux is to be measured. Flux density is an example of a vector which depends upon an area, and is sometimes called a bivector. The notion of two vectors involved in the term bivector may 30 VECTOR CALCULUS be avoided by the term cycle, or the term feuille. It is also called an axial vector, in opposition to the ordinary vectors, called polar vectors. The term axial is applicable in the sense that it is the axis or normal of a portion of a surface. The portion (feuille, cycle) of the surface is traversed in the positive direction in going around its boundary, that is, with the surface on the left-hand. If the direction of the axial vector is reversed, we also traverse the area attached in the reverse direction, so that in this sense the axial vector may be regarded as invariant for such change while the polar vector would not be invariant. The distinction is not of much importance. The important idea is that of areal integration for the flux density or any other so-called axial vector, while the polar vector is sub- ject only to linear integration. We meet the distinction in the difference below between the induction vectors and the intensity vectors. (9) Energy Density Current. R [TOL -2 ? 7 - 2 ]. When an energy density has the idea of velocity attached to it, it becomes a vector with the given dimensions. In such case we consider it as of the nature of a flux density. (10) Energy Current. 2 [$QT~ 2 ]. If a vector of energy density current is multiplied by an area we arrive at an energy current. (11) Electric Density Current. J [SL^T- 1 ]. A number of moving electrons will determine an average density per square centimeter across the line of flow, and the product of this into a velocity will give an electric density current. To this must also be added the time rate of change of electric induction, which is of the same dimensions, and counts as an electric density current. (12) Electric Current. C [97 1-1 ]. The unit is the ampere = 3-10 9 e.s. units = 10 _1 e.m. units. This is the product of an electric density current by an area. VECTOR FIELDS 31 (13) Magnetic Density Current. G [$Ir 2 T- 1 }. Though there is usually no meaning to a moving mass of magnetism, nevertheless, the time rate of change of magnetic induction must be considered to be a current, similar to electric current density. (14) Magnetic Current. K [^T' 1 ]. The unit is the heavy side = 1 e.m. unit = 3 • 10 10 e.s. units. In the phenom- ena of magnetic leakage we have a real example of what may be called magnetic current. Both electric current and magnetic current may also be scalars. For instance, if the corresponding flux densities are integrated over a given surface the resulting scalar values would give the rate at which the electricity or the magnetism is passing through the surface per second. In such case the symbols should be changed to corresponding Roman capitals. (15) Electric Intensity. E fMr 1 ! 1 " 1 ]. When an electric charge is present in any portion of space, there is at each point of space a vector of a field called the field of electric intensity. The same situation happens when lines of magnetic induction are moving through space with a given velocity. The electric intensity will be perpendicular to both the line of magnetic induction and to the velocity it has, and equal to the product of their intensities by the sine of their angle. The electric intensity is of the nature of a polar vector and its flux, or surface integral over any surface has no meaning. Its line integral along any given path, however, is called the difference of voltage between the two points at the ends of the path, for that given path. The unit of voltage is the volt = J • 10~ 2 e.s. units = 10 8 e.m. units. The symbol for voltage is V [$T~ 1 ]. Its dimensions are the same as for scalar electric potential, or magnetic current. 32 VECTOR CALCULUS (16) Electric Induction. D [QL~ 2 ]. The unit is the line = 3-10 9 e.s. units — 10 -1 e.m. units. This vector usually has the same direction as electric intensity, but in non- isotropic media, such as crystals, the directions do not agree. It is a linear function of the intensity, however, ordinarily indicated by D = k(E) where k is the symbol for a linear operator which converts vectors into vectors, called here the permittivity, [0^> -1 Z _1 T], measurable in farads per centimeter. In isotropic media k is a mere numerical multiplier with the proper dimensions, which are essential to the formulae, and should not be neglected even when k = 1. The flux is measured in coulombs. (17) Magnetic Intensity. H [eL" 1 ? 7 " 1 ]. The field due to the poles of permanent magnets, or to a direct current traversing a wire, is a field of magnetic intensity. In case we have moving lines of electric induction, there is a field of magnetic intensity. It is of a polar character, and its flux through a surface has no meaning. The line integral between two points, however, is called the gilbertage between the points along the given path, the unit being the gilbert = 1 e.m. unit = 3 • 10 10 e.s. units. The symbol is N [GT- 1 ]' Its dimensions are the same as those of scalar magnetic potential, or electric current. (18) Magnetic Induction. B [$L~ 2 ]. The unit is the gauss = 1 e.m. unit = 3 • 10 10 e.s. units. The direction is usually the same as that of the intensity, but in any case is given by a linear vector operator so that we have B-m(H) where \x is the inductivity, [^>0 -1 Z _1 T], measurable in henrys per centimeter. The flux is measured in maxwells. VPPf VECTOR FIELDS 33 (19) Vector Potential of Electric Induction. T [eZ -1 ]. A vector field may be related to another vector field in a certain manner to be described later, such that the first can be called the vector potential of the other. (20) Vector Potential of Magnetic Induction. ^ [M -1 ]. This is derivable from a field of magnetic induction. This and the preceding are line-integrable. (21) Hertzian Vectors. 9, <£. These are line integrals of the preceding two, and are of a vector nature. 5. Vector Lines. If we start at a given point of a vector field and consider the vector of the field at that point to be the tangent to a curve passing through the point, the field will determine a set of curves called a congruence, since there will be a two-fold infinity of curves, which will at every point have the vector of the field as tangent. If the field is represented by a, a function of p, the vector to a point of the field, then the differential equation of these lines of the congruence will be dp = adt, where dt is a differential parameter. From this we can determine the equation of the lines of the congruence, in- volving an arbitrary vector, which, however, will not have more than two essential constants. For instance, if the field is given by a = p, then dp = pdt, and p = ae l , where a is a constant unit vector. The lines are, in this case, the rays emanating from the origin. The lines can be constructed approximately by starting at any given point, thence following the vector of the field for a small distance, from the point so reached following the new vector of the field a small distance, and so proceed- ing as far as necessary. This will trace approximately a vector line. Usually the curves are unique, for if the field is monodromic at all points, or at points in general, the 34 VECTOR CALCULUS curves must be uniquely determined as there will be at any point but one direction to follow. Two vector lines may evidently be tangent at some point, but in a monodromic field they cannot intersect, except at points where the in- tensity of the field is zero, for vectors of zero intensity are of indeterminate direction. Such points of intersection are singular points of the field, and their study is of high importance, not only mathematically but for applications. In the example above the origin is evidently a singular point, for at the origin a = 0, and its direction is indetermi- nate. 6. Vector Surfaces, Vector Tubes. In the vector field we may select a set of points that lie upon a given curve and from each point draw the vector line. All such vector lines will lie upon a surface called a vector surface, which in case the given curve is closed, forming a loop, is further particularized as a vector tube. It is evident that the vector lines are the characteristics of the differential equation dp = adt, which in rectangular coordinates would be equivalent to the equations dx _dy _ dz X ~ Y~ Z' In case these equations are combined so as to give a single exact equation, the integral will (since it must con- tain a single arbitrary constant) be the equation of a family of vector surfaces. The vector lines are the intersections of two such families of vector surfaces. The two families may be chosen of course in infinitely many different ways. Usually, however, as in Meteorology, those surfaces are chosen which have some significance. When a vector tube becomes infinitesimal its limit is a vector line. 7. Isogons. If we locate the points at which a has the VECTOR FIELDS 35 same direction, they determine a locus called an isogon for the field. For instance, we might locate on a weather map all the points which have the same direction of the wind. If isogons are constructed in any way it becomes a simple matter to draw the vector lines of the field. Machines for the use of meteorologists intended to mark the isogons have been invented and are in use.* As an instance con- sider the vector field a = (2x, 2y, — z). An isogon with the points at which a has the direction whose cosines are /, m, n is given by the equations 2x : 2y : — z = I : m : n or 2x = It, 2y = mt, z = — nt. It follows that the vector to any point of this isogon is given by p = t(l, m, n) - (0, 0, 3nt). That is to say, to draw the vector p to any point of the isogon we draw a ray from the origin in the direction given, then from its outer end draw a parallel to the Z direction backward three times the length of the Z projection of the segment of the ray. The points so determined will evi- dently lie on straight lines in the same plane as the ray and its projection on the XY plane, with a negative slope twice the positive slope of the ray. The tangents of the vector lines passing through the points of the isogon will then be parallel to the ray itself. The vector lines are drawn ap- proximately by drawing short segments along the isogon parallel to its corresponding ray, and selecting points such that these short segments will make continuous lines in *Sandstrdm: Annalen der Hydrographie und Maritimen Meteor- ologie (1909), no. 6, pp. 242 et.seq. Bjerknes: Dynamic Meteorology. See plates, p. 50. 36 VECTOR CALCULUS passing to adjacent isogons. The figure illustrates the method. All the vector lines are found by rotating the figure about the X axis 180°, and then rotating the figure so produced about the Z axis through all angles. Fig. 3. 8. Singularities. It is evident in the example preceding that there are in the figure two lines which are different from the other vector lines, namely, the Z axis and the line which is in the XY plane. Corresponding to the latter would be an infinity of lines in the XY plane passing through the origin. These lines are peculiar in that the other vector lines are asymptotic to them, while they are themselves vector lines of the field. A method of studying the vector lines in the entire extent of the plane in which they lie was used by Poincare. It consists in placing a sphere tangent VECTOR FIELDS 37 to the plane at the origin. Lines are then drawn from the center of the sphere to every point of the plane, thus giving two points on the sphere, one on the hemisphere next the plane and one diametrically opposite on the hemisphere away from the plane. The points at infinity in the plane correspond to the equator or great circle parallel to the plane. In this representation every algebraic curve in the plane gives a closed curve or cycle on the sphere. In the present case, the axes in the plane give two perpendicular great circles on the sphere, and the vector lines will be loops tangent to these great circles at points where they cross the equator. These loops will form in the four Junes of the sphere a system of closed curves which Poincare calls a topographical system. The equator evidently belongs to the system, being the limit of the loops as they grow nar- rower. The. two great circles corresponding to the axes also belong to the system, being the limits of the loops as they grow larger. If a point describes a vector line its projection on the sphere will describe a loop, and could never leave the lune in which the projection is situated. The points of tangency are called nodes', the points which represent the origin, and through which only the singular vector lines pass, are called fames. 9. Singular Points. The simplest singular lines depend upon the singular points and these are found comparatively simply. The singular points occur where o" = or a —• oo . Since we may multiply the components of a by any ex- pressions and still have the lines of the field the same, we may equally suppose that the components of a are reduced to as low terms as possible by the exclusion of common factors of all of them. We will consider first the singular 38 VECTOR CALCULUS points for fields in space, then those cases which have lines every point of which is a singular point, which will include the cases of plane fields, since these latter may be considered to represent the fields produced by moving the plane field parallel to itself. The classification given by Poincare is as follows. (1) Node. At a node there may be many directions in which vector lines leave the point. An example is a = p. At the origin, it is easy to see, a = 0, and it is not possible to start at the origin and follow any definite direction. In fact the vector lines are evidently the rays from the origin in all directions. There is no other singular point at a finite distance. If, however, we consider all the rays in any one plane, and for this plane construct the sphere of projection, we see that the lines correspond to great circles on the sphere which all pass through the origin and the point diametrically opposite to it. This ideal point may be considered to be another node, so that all the vector lines run from node to node, in this case. Every vector line which does not terminate in a node is a spiral or a cycle. (2) Faux. From a faux* there runs an infinity of vector lines which are all on one surface, and a single isolated vector line which intersects the surface at the faux. The surface is a singular surface since every vector line in it through the faux is a singular line. The singular surface is approached asymptotically by all the vector lines not singular. An example is given by a = (x, y, — z). The vector lines are to be found by drawing all equilateral hyperbolas in the four quadrants of the ZX plane, and then * Poincare uses the term col, meaning mountain pass, for which faux is Latin. / VECTOR FIELDS 39 rotating this set of lines about the Z axis. Evidently all rays in the XY plane from the origin are singular lines, as well as the Z axis. Where fauces occur the singular lines through them are asymptotes for the nonsingular lines. If Fig. 4. we consider any plane through the Z axis, the system of equilateral hyperbolas will project onto its sphere as cycles tangent on the equator to the great circles which repre- sent the singular lines in that plane. From this point of view we really should consider the two rays of the Z axis as separate from each other, so that the upper part of the Z axis and the singular ray perpendicular to it, running in the same general direction as the other vector lines, would con- stitute a vector line with a discontinuity of direction, or with an angle. Such a vector line to which the others are tangent at points at infinity only is a boundary line in the sense that on one side we have infinitely many vector lines which form cycles (in the sense defined) while on the other sides we have vector lines which belong to different sys- tems of cycles. 40 VECTOR CALCULUS A simple case of this example might arise in the inward flow of air over a level plane, with an ascending motion which increased as the air approached a given vertical line, becoming asymptotic to this vertical line. In fact, a small fire in the center of a circular tent open at the bottom for a small distance and at the vertex, would give a motion to the smoke closely approximating to that described. A singular line from a faux runs to a node or else is a spiral or part of a cycle which returns to the faux. An example that shows both preceding types is the field a = (x 2 + y 2 — 1, bxy — 5, mz). In the X Y plane the singular points are at infinity as follows : A at the negative end of the X axis, and B at the positive end, both fauces; C at the end of the ray whose direction is tan -1 2, in the first quadrant, D at the end of the ray of direction tan -1 2 in the third quadrant; E at the end of the VECTOR FIELDS 41 ray of direction tan -1 — 2 in the fourth quadrant; and F at the end of the ray of tan -1 — 2 in the second quadrant, these four being nodes. Vector lines run from E to D separated from the rest of the plane by an asymptotic division line from B to D; from C to D on the other side of this division line, separated from the third portion of the plane by an asymptotic division line from C to A ; and from C to F in the third portion of the plane. The figure shows the typical lines of the field. (3) Focus. At a focus the vector lines wind in asymp- totically, either like spirals wound towards the vertex of a spindle produced by rotating a curve about one of its tangents, one vector line passing through the focus, or they are like spirals wound around a cone towards the Fig. 6. vertex. As an example o- = (x+ y, y - x, z). The Z axis is a single singular line through the origin, which is a singular point, a focus in this case. The XY plane contains vector lines which are logarithmic spirals wound in towards the origin. The other vector lines are spirals 42 VECTOR CALCULUS wound on cones of revolution, their projections on XY being the logarithmic spirals. By changing z to az we would have different surfaces depending upon whether 1 < a. a< 1 or In case a spiral winds in onto a cycle, the successive turns approaching the cycle asymptotically, the cycle is called a limit cycle. In this example the line at infinity in the X Y plane, or the corresponding equator on its sphere, is a limit cycle. It is clear that the spirals on the cones wind outward also towards the lines at infinity as limit cycles. From this example it is plain that vector lines which are spiral may start asymptotically from a focus and be bounded by a limit cycle. The limit cycle thus divides the plane or the surface upon which they lie into two mutually exclusive regions. Vector lines may also start from a limit cycle and proceed to another limit cycle. As an example of vector lines of both kinds consider the field Fig. 7. a = ( r 2 _ 1, r 2 + lf mz)f where the first component is in the direction of a ray in the XY plane from the origin, the second perpendicular to VECTOR FIELDS 43 this in the XY plane, and the third is parallel to the Z axis. The vector lines in the singular plane, the XY plane, are spirals with the origin as a focus for one set, which wind around the focus negatively and have the unit circle as a limit cycle, while another set wind around the unit circle in the opposite direction, having the line at infinity as a limit cycle. The polar equation of the first set is r~ l — r An example with all the preceding kinds of singularities is the field Fig. 8. a = ( [r 2 - l)(r - 9)], (r 2 - 2r cos 9 - 8), mz) with directions for the components as in the preceding example. The singular points are the origin, a focus; the point A (r = 3, = + cos -1 §), a node; the point B (r = 3, 6 = — cos -1 J), a faux. The line at infinity is a limit cycle, as well as the circle r = 1, which is also a vector line. The circle r = 3 is a vector line which is a cycle, 4 44 VECTOR CALCULUS starting at the faux, passing through the node and returning to the faux. The vector lines are of three types, the first being spirals that wind asymptotically around the focus, out to the unit circle as limit cycle; the second start at the node A and wind in on the unit circle as limit cycle; the third start at the node A and wind out to the line at in- finity as unit cycle. The second set dip down towards the faux. The exceptional vector lines are the line at infinity, the unit circle, both being limit cycles; the circle of radius 3; a vector line which on the one side starts at the faux B winding in on the unit circle, and on the other side starts at the faux B winding outward to the line at infinity as limit cycle. The last two are asymptotic division lines of the regions. The figure exhibits the typical curves. (4) Faux-Focus. This type of singular point has passing through it a singular surface which contains an infinity of spirals having the point as focus, while an isolated vector line passes through the point and the surface. No other surfaces through the vector lines approach the point. An instance is the field a- = (x, y, — z). The Z axis is the isolated singular line, while the XY plane is the singular plane. In it there is an infinity of spirals with the origin as focus and the line at infinity as limit cycle. All other vector lines lie on the surfaces rz = const. These do not approach the origin. (5) Center. At a center there is a vector line passing through the singular point, and not passing through this singular line there is a singular surface, with a set of loops or cycles surrounding the center, and shrinking upon it. There is also a set of surfaces surrounding the isolated singular line like a set of sheaths, on each of which there are vector lines winding around helically on it with a decreasing VECTOR FIELDS 45 Fig. 9. pitch as they approach the singular surface, which they therefore approach asymptotically. As an instance we have the field a = (y, - x, z). The Z axis is the singular isolated vector line, the XY plane the singular surface, circles concentric to the origin the singular vector lines in it, and the other vector lines lie on circular cylinders about the Z axis, approaching the XY plane asymptotically. The method of determining the character of a singular point will be considered later in connection with the study of the linear vector operator. A singular point at infinity is either a node or a faux. 10. Singular Lines. Singularities may not occur alone but may be distributed on lines every point of which is a singular point. This will evidently occur when cr = gives three surfaces which intersect in a single line. The dif- ferent types may be arrived at by considering the line of singularities to be straight, and the surfaces of the vector lines with the points of the singular line as singularities to be planes, -for the whole problem of the character of the singularities is a problem of analysis situs, and the deforma- tion will not change the character. The types are then as follows : (1) Line of Nodes. Every point of the singular line is a node. A simple example is a = (x, y, 0). The vector lines are all rays passing through the Z axis and parallel to the XY plane. 46 VECTOR CALCULUS (2) Line of Fauces. There are two singular vector lines through each point of the singular line. As an instance a = (x, — y, 0). The lines through the Z axis parallel to the X and the Y axes are singular, all other vector lines lying on hyperbolic cylinders. (3) Line of Foci. The points of the singular line are approached asymptotically by spirals. As an instance <t = (x + y, y — x, 0). The vector lines are logarimithic spirals in planes parallel to the XY plane, wound around the Z axis which is the singular line. (4) Line of Centers. A simple case is a — (y, — x, 0). The vector lines are the Z axis and all circles with it as axis. 11. Singularities at Infinity. The character of these is determined by transforming the components of a so as to bring the regions at infinity into the finite parts of the space we are considering. The asymptotic lines will then have in the transformed space nodes at which the lines are tangent to the asymptotic line. 12. General Characters. The problem of the character of a vector field so far as it depends upon the vector lines and their singularities is of great importance. Its general resolution is due to Poincare. In a series of memoirs in the Journal des Mathematiques* he investigated the qualitative character of the curves which represent the characteristics of differential equations, particularly with the intention of bringing the entire set of integral curves into view at once. Other studies of differential equations usually relate to the character of the functions defined at single points and in their vicinities. The chief difficulty of the more general study is to ascertain the limit cycles. These with the asymptotic division lines separate the field into independent regions. * Ser. (3) 7 (1881), p. 375; ser. (3) 8 (1882), p. 251; ser. (4) 1 (1885), p. 167. Also Takeo Wado, Mem. Coll. Sci. Tokyo, 2 (1917) 151. VECTOR FIELDS 47 The asymptotic division lines appear on meteorological maps as lines on the surface of the earth towards which, or away from which, the air is moving. They are called in the two cases lines of convergence, or lines of divergence, respectively. If a division line of this type starts at a node the node may be a point of convergence or a point of divergence. The line will then have the same character. The node in other fields, such as electric or magnetic or heat flow, is a source or a sink. If a division line starts from a faux, the latter is often called a neutral point. A focus may be also a point of convergence or point of divergence. In the case of a singular line consisting of foci, the singular line may be a line of convergence or of divergence; in the first case, for instance, the singular line is the core of the anti- cyclone, in the latter case, the core of the cyclone. The limit cycles which are not at infinity are division lines which enclose areas that remain isolated in the field. Such phenomena as the eye of the cyclone illustrate the oc- currence of limit cycles in natural phenomena. The limit cycle may be a line of convergence or a line of divergence, the air in the first case flowing into the line asymptotically from both inside and outside, with the focus serving as a source, and in the other case with conditions reversed. The practical handling of these problems in meteorological work depends usually upon the isogonal lines: the lines which are loci of equidirected tangents of the vector lines of the field. These are drawn and the infinitesimal tan- gents drawn across them. The filling in of the vector lines is then a matter of draughtsmanship. The isogonal lines will themselves have singularities and these will enable one to determine somewhat the singularities of the vector lines themselves. Since the unit vector in the direction of a is constant along an isogon it is evident that 48 VECTOR CALCULUS the only change in a along an isogon is in its intensity, that is, a keeps the same direction, and its differential is therefore a multiple of a, that is, the isogons have for their differential equation da = adt. Consequently, when a = or a = <x> the isogon will have a singular point. It does not follow, however, that all the singular points of the isogons will appear as singular points such as are described above for the vector lines. When the differential equation of the isogons is reduced to the standard form dp = rdu we shall see later that r will be a linear vector function of a, and that a linear vector function may have zero directions, so that <pa — 0, without a = 0. Some of the phenomena that may happen are the following, from Bjerknes' Dynamic Meteorology and Hydrography. See his plates 42a, 426. 1. Node of Isogons. These may be positive, in which case the directions of the tangents of the vector lines will increase (that is, the tangent will turn positively) as succes- sive isogons are taken in a positive rotation about the node, or may be negative in the reverse case. The positive node of the isogon will then correspond to a node, a focus, or a center of the vector lines. The negative node of the isogon will correspond to a faux of the vector lines. If the isogons are parallel, having, therefore, a node at infinity in either of their directions, the vector lines may have asymptotic division lines running in the same direc- tion, or they may have lines of inflexion parallel to the isogons. 2. Center of Isogons. When the isogons are cycles they may correspond to very complicated forms of the vector VECTOR FIELDS 40 lines. Several of these are to be found in a paper by Sand- strom, Annalen der Hydrographie und maritimen Meteor- ologie, vol. 37 (1909), p. 242, Uber die Bewegung der Flussigkeiten. EXERCISES* * To be solved graphically as far as possible. 1. A translation field is given by a- = (at, bt, ct), what are the vector lines, the isogons, and the singularities? 2. A rotation field is given by a = (mz — ny, nx — Iz, ly — mx), what are the isogons, singularities, and vector lines? 3. A field of deformation proportional to the distance in one direction is given by a = {ax, 0, 0). Determine the field. 4. A general field of linear deformation is given by o- = (ax + by + cz, fx + gy + hz, kx + ly + tnz) . determine the various kinds of fields this may represent according to the different possible cases. 5. Consider the quadratic field* a = (x 2 — y 2 — z 2 , 2xy, 2xz). 6. Consider the quadratic field a = (xy — xz, yz — yx, zx — zy). 7. What are the lines of flow when the motion is stationary in a rotating fluid contained in a cylindrical vessel with vertical axis of rotation? 8. Consider the various fields a = (ay -\- x, y — ax, b) for different values of a, which is the tangent of the angle between the curves and their polar radii. What happens in the successive diagrams to the isogons, to the curves? 9. Consider the various fieldsf a = (l,f(r — a), b) where r is the polar radius in the XY plane, a is constant, and / takes the various forms f(x) = x, x 2 , x 3 , x 112 , x 113 , x~ l , x~ 2 , e x , log x, sin x, tan x. 10. Consider the forms a = (1, f(air sin r), b) where j(x) = sin x, cos x, tan x. 11. In various electrical texts, such as Maxwell, Electricity and Magnetism, and others, there will be found plates showing the lines of various fields. Discuss these. Also, the meteorological maps in Bjerknes' Dynamic Meteorology, referred to fibove. * See Hitchcock, Proc. Amer. Acad. Arts and Sci., 12 (1917), No. 7, pp. 372-454. f See Sandstrom cited above. 50 VECTOR CALCULUS 12. In a funnel-shaped, vortex of a water-spout the spout may be considered to be made up of twisted funnels, one inside another, the space between the surfaces being a vortex tube. In the Cottage City water-spout, Aug. 19, 1896, the equation of the outside funnel may be taken to be (z 2 + y*)z = 3600. In this x, y are measured horizontally in meters from the axis of the tubes, and z is measured vertically downwards from the cloud base, which is 1100 meters above the ground. The inner surfaces have the same equation save that instead of 3600 on the right we have 3600/(1. 60 10) 2n ; that is, at any level, the radius of a surface bounding a tube is found from the preceding radius at the same level by dividing by the number whose logarithm (base 10) is 0.20546. From meteoro- logical theory the velocity of the wind on any surface is given by <r = (Cr, Crz, - 2Cz) where the first component is the horizontal radial component, the second is the tangential, and the third is the vertical component. C varies for the different surfaces, and is found by multiplying the value for the outside surface by the square of the number 1.6010. In Bige- low's Atmospheric Radiation, etc., p. 200 et seq., is to be found a set of tables for the various values from these data for different levels. Char- acterize the vortex field of the water-spout. 13. For a dumb-bell-shaped water-spout, likewise, the funnels have the equation (x 2 + y 2 ) sin az — const/A where A varies from surface to surface just as C in the preceding problem. The velocity is given by o- = (— Aar cos az, Aar sin az, 2A sin az), the directions being horizontal radial, tangential and vertical. For the St. Louis tornado, May 27, 1896, the following data are given. Cloud base 1200 meters above the ground, divided into 121 parts called degrees, the ground thus being at 60°, and az being in degrees. The values of A are for the successive funnels 0.1573, 0.4052, 1.0437, 2.6883, 6.9247, 17.837. Characterize the vector Ikies of this vortex field. 14. In the treatise on The Sun's Radiation, Bigelow gives the follow- ing data for a funnel-shaped vortex r 2 z = 6400000/C W=wind--from9f PLATE I PLATE II VECTOR FIELDS 51 at 500 kilometers z = 500, r = 60474, 26287, 11513, 5023, 2192, 956. a (Km/sec) = (Cr, Crz, - 2Cz). Calculate for z = 0, 500, 1000, 2000, 5000, 10000, 20000, 30000, 40000, 50000. The results of the calculations give a vortex field agreeing with Hale's observations. The vector lines in the last three problems lie on the funnel surfaces, being traced out in fact by a radius rotating about the axis of the vortex, and advancing along the axis according to the law 2d = - z + C for the funnel, 20 = az + C for the dumb-bell. 15. Study the lines on the plates, which represent on the first plate the isogons for wind velocities, on the second plate the corresponding characteristic lines of wind flow. The date was evening of Jan. 9, 1908. European and American systems of numbering directions are shown in the margin of plate 1. See Sandstrom's paper cited above. 13. Congruences. We still have to consider the relations of the various vector lines to each other, noticing that the vector lines constitute geometrically a congruence, that is, a two-parameter system of curves in space. The con- sideration of these matters, however, will have to be post- poned to a later chapter. CHAPTER IV ADDITION OF VECTORS 1 . Sum of Vectors. Geometrically, the sum of two or more vectors is found by choosing any one of them as the first, from the terminal point of the first constructing the second (any other), from the terminal point of this con- structing the third (any of those left) and so proceeding till all have been successively joined to form a polygon in space with the exception of a final side. If now this last side is constructed by drawing a vector from the initial point of the first to the terminal point of the last, the vector so drawn is called the sum of the several vectors. In case the polygon is already closed the sum is a zero vector. When the sum of two vectors is zero they are said to be opposite, and subtraction of a vector consists in adding its opposite. It is evident from the definition that we presuppose a space in which the operations can be effectively carried out. For instance, if the space were curved like a sphere, and the sum of two vectors is found, it would evidently be different according to which is chosen as the first. The study of vector addition in such higher spaces has, however, been con- sidered. Encyclopedic des sciences mathematiques, Tome IV, Vol. 2. 2. Algebraic Sum. In order to define the sum without reference to space, it is necessary to consider the hyper- numbers that are the algebraic representatives of the geometric vectors. We must indeed start with a given set of hypernumbers, which are the basis of the system of hypernumbers we in- tend to study. These are sometimes called imaginaries, because they are analogous to V— 1. In the case of three- 52 ADDITION OF VECTORS 53 dimensional space there are three such hypernumbers in the basis. We combine in thought a numerical value with each of these, the field or domain from which these numeri- cal values are chosen being of great importance. For in- stance, we may limit our numbers to the domain of integers, the domain of rationals, the domain of reals, or to other more complicated domains, such as certain algebraic fields. We then consider all the multiplexes we can form by put- ting together into a single entity several of the hypernum- bers just formed, as for instance, we would have in three- dimensional space such a compound as p = («1, 7/e 2 , Z€ 3 ). Since we are now using the base hypernumbers e it is no longer necessary to use the parentheses nor to pay attention to the order of the terms. We drop the use of the comma, however, and substitute the + sign, so that we would now write p = X€i + 2/€ 2 + 2€ 3 . We may now easily define the algebraic sum of several hypernumbers corresponding to vectors by the formula Pi = Xi€i + y { €2 + Zi*z, [ i = 1, 2, • • • m, ]T Pi = 2£i€i + 2^€ 2 + 2Zi€ 3 . i = 1 This definition of course includes subtraction as a special case. It is clear from this definition that to correspond to the geometric definition, it is necessary that the units e corre- spond to three chosen unit vectors of the space under con- sideration. They need not be orthogonal, however. The coefficients of the e are then the oblique or rectangular coordinates of the point which terminates the vector if it starts at the origin. 54 VECTOR CALCULUS 3. Change of Basis. We may define all the hyper- numbers of the system in terms of a new set linearly related to the original set. For instance, if we write €1 = duOti + ai2«2 + Ol3«3, € 2 = CiziOLl + a22«2 "T" 023«3> €3 = a 3 iai + a 32 a 2 + «33«3, then p becomes P = (a n x + a n y + a n z)ai + (a u x + a 22 y + a 32 z)a 2 + (a n x -f a 2z y + a 33 z)a 3 . It is evident then that if we transform the e's by a non- singular linear homogeneous transformation, the coeffi- cients of the new basis hypernumbers, a, are the transforms of the original coefficients under the contragredient trans- formation. Inasmuch as the transformation is linear, the transform of a sum will be the sum of the transforms of the terms of the original sum. The transformation as a geometrical process is equivalent to changing the axes. This process evidently gives us a new triple, but must be considered not to give us a new hypernumber nor a new vector. Indeed, a vector cannot be defined by a triple of numbers alone. There is also either explicitly stated or else implicitly understood to be a basis, or on the geometric side a definite set of axes such that the triple gives the components of the vector along these axes. It is evident that the success of any system of vector calculus must then depend upon the choice of modes of combination which are not affected by the change from one basis to another. This is the case with addition as we have defined it. We assume that we may express any vector or hypernumber in terms of any basis we like, and usually the basis will not appear. If the transformation is such as to leave the angles be- ADDITION OF VECTORS 55 tween ei, e 2 , e 3 the same as those between a\, a 2 , a 3 , the second trihedral being substantially the same as the first rotated into a new position, with the lengths in each case remaining units, then the transformation is called orthog- onal. We may define an orthogonal transformation algebra- ically as one such that if followed by the contragredient transformation the original basis is restored. 4. Differential of a Vector. If we consider two points at a small distance apart, the vector to one being p, to the other p', and the vector from the first to the second, Ap = p' — p, where Ap = As-e, e being a unit vector in the direction of the difference, we may then let one point ap- proach the other so that in the limit e takes a definite posi- tion, say a, and we may write ds for As, and call the result the differential of p for the given range over which the p f runs. In the hypernumbers we likewise arrive at a hyper- number dp = dxei -f- dye?, + dzez, where now ds is a linear homogeneous irrational function of dx, dy, dz, which = V (dx 2 + dy 2 + dz 2 ) in case e ly e 2 , e 3 form a trirectangular system of units. The quotient dpjdt is the velocity at the point if t repre- sents the time. The unit vector a: is the unit tangent for a curve. We generally represent the principal normal and the binormal by jS, 7 respectively. When p is given as dependent on a single variable parameter, as t for instance, then the ends of p may describe a curve. We may have in the algebraic form the coordinates of p alone dependent upon the parameter, or we may have both the coordinates and the basis dependent upon t. For instance, we may ex- press p in terms of ei, e 2 , e 3 which are not dependent upon t but represent fixed directions geometrically, or we may express p in terms of three hypernumbers as w, r, J* which 56 VECTOR CALCULUS themselves vary with t, such as the moving axes of a system in space. In relativity theories the latter method of repre- sentation plays an important part. 5. Integral of a Vector. If we add together n vectors and divide the result by n we have the mean of the n vectors, which may be denoted by p. If we select an infinite number of vectors and find the limit of their sum after multiplication by dt, the differential of the parameter by which they are expressed, such limit is called the integral of the vector expressed in terms of t, and if we give t two definite values in the integral and subtract one result from the other, the difference is the integral of the vector from the first value of t to the second. More generally, if we multiply a series of vectors, infinite in number, by a corre- sponding series of differentials, and find the limit of the sum of the results, such limit, when it exists, is called the integral of the series. In integration, as in differentiation, the usual difficulties met in analysis may appear, but as they are properly difficulties due to the numerical system and not to the hypernumbers, we will suppose that the reader is familiar with the methods of handling them. The mean in the case of a vector which has an infinite sequence of values is the quotient of the integral taken on some set of differentials, divided by the integral of the set of differentials itself. The examples will illustrate the use of the mean. EXAMPLES (1) The centroid of an arc, an area, or a volume is found by integrating the vector p itself multiplied by the dif- ferential of the arc, ds, or of the surface, du, or of the volume dv. The integral is then divided by the length of the arc, the area of the surface, or the volume. That is - Sheets ffpdu • fffpdv m P — — , or - — or — b— a A V ADDITION OF VECTORS 57 (2) An example of average velocity \s found in the following (Bjerknes, Dynamic Meteorology, Part II, page 14) obser- vations of a small balloon. 2 = Ht. in Meters Az Direction Velocity (w/sec.) Products 77 680 960 1240 1530 1810 2090 2430 2730 3040 3400 3710 4030 4400 603 280 280 290 280 280 340 300 310 360 310 320 370 S. 50° E. S. 57° E. S. 36° E. S. 28° W. S. 2°W. S. 2°W. S. 35° W. S. 53° W. S. 69° W. S. 55° W. S. 53° W. S. 58° W. S. 37° W. 3.4 4.0 5.3 1.5 1.8 2.0 1.5 1.8 1.8 3.0 2.8 4.4 10.2 2050 1120 1484 435 504 560 510 540 558 1080 868 1408 3773 To average the velocities we notice that on the assump- tion that the upward velocity was uniform the distances vertically can be used to measure the time. We therefore multiply each velocity by the difference of elevations corresponding, the products being set in the last column. These numbers are then taken as the lengths of the vectors whose directions are given by the third column. The sum of these is found graphically, and divided by the total difference of distance upward, that is, 4323. In the same manner we can find graphically the averages for each 1000 meters of ascent. We may now make a new table in order to find other important data, as follows : Height . Pressure (ra-bars) Dens, (ton/w 3 ) Veloc. . Spec. Mo- mentum (ton/ra 2 sec.) 4000 3000 2000 1000 75 622 705 797 899 1003 0.00083 0.00092 0.00102 0.00112 3.8 1.6 • 2.4 3.7 0.0032 0.0015 0.0025 0.0041 ;,s VECTOR CALCULUS We now find the average velocity between the 1000 m-bar, the 900 ra-bar, the 800 m-bar, the 700 ra-bar, and the 600 ra-bar. The direction is commonly indicated by the in- tegers from to 63 inclusive, the entire circle being divided into 64 parts, each of 5f°. East is 0, North is 16, NW. is 24, etc. The following table is found. Pressure Height Spec. Vol. (m 3 /Ton) Direction Veloc. Spec. Mo- mentum 600 700 800 ' 900 1000 1002.6 4274 3057 1970 989 99 76 1217 1087 981 890 890 8 7 20 25 25 5.2 1.7 2.4 3.7 3.4 0.0043 0.0016 0.0024 0.0042 0.0040 Of course, specific momenta should be averaged like veloc- ities but usually owing to the rough measurements it is sufficient to find specific momenta from the average velocities. ADDITION OF VECTORS 59 EXERCISES 1. Average as above the following observations taken at places mentioned (Bjerknes, p. 20), July 25, 1907, at 7 a.m. Greenwich time. Isobar Dyn. Ht. Az Direction Veloc. 100 200 300 400 500 600 700 800 900 1000 1001.2 16374 11947 9320 7301 5648 4240 3020 1938 975 107 98 4427 2627 2019 1653 1408 1220 1082 963 867 9 10 18 19 8 5 4 4 36 35 4.7 3.2 3.4 3.3 2.6 2.5 2.5 1.4 4.5 4.5 Uccle, Lat. 50° 48' Long. 4° 22' 100 200 300 400 500 600 700 800 900 955.9 16238 11817 9240 7248 5626 4244 3038 1955 977 471 4421 2577 1992 1622 1382 1206 1083 978 506 3 6 7 2 3 2 62 4 30 10.0 6.5 7.6 10.2 6.7 6.8 5.3 0.6 2.1 Zurich, Lat. 47° 23' Long. 8° 33' 200 300 400 500 600 700 800 900 1000 11890 9241 7240 5643 4196 2991 1927 981 118 17 2649 2001 1597 1447 1205 1064 946 863 101 59 57 58 55 49 41 38 56 55 9.2 10.5 8.8 8.0 2.9 2.9 1.9 4.3 3.4 Hamburg, Lat. 53° 33' Long. 9° 59' GO VECTOR CALCULUS 2. If the direction of the wind is registered every hour how is the average direction found? Find the average for the following observa- tions. Station Pikes Peak Vienna Mauritius Cordoba S Orkneys Elev 4308 m. 26 m. 15 m. 437 m. 25 m Summer Winter Dec-Feb. Winter Summer Time Vel. Az. Vel. Az. Vel. Az. Vel. Az. Vel. Az. a.m 0.84 1.34 1.46 1.05 0.43 0.66 1.03 100° 83 71 57 12 279 262 0.47 0.56 0.42 0.33 0.22 0.17 0.36 62 61 59 57 46 303 257 1.00 1.30 1.30 1.00 1.10 1.80 2.40 100 97 98 119 241 312 326 0.94 1.06 1.44 2.03 2.17 0.50 2.78 115 111 121 132 136 252 314 0.52 0.52 0.51 0.51 0.52 0.53 0.54 70 2 51 4 30 6 5 8 343 10 ?m 12 noon 255 14 1.09 256 0.58 242 1.60 332 3.56 315 0.54 245 16 0.95 253 0.64 232 1.30 304 3.36 305 0.54 42 18 0.74 247 0.47 223 0.20 10 1.75 299 0.53 35 20 0.49 47 0.14 186 0.90 101 0.72 44 0.53 50 22 0.36 153 0.25 72 1.00 102 0.89 128 0.52 60 Bigelow, Atmospheric Circulation, etc., pp. 313-315. 3. The following table gives the mean magnetic deflecting vectors, in four zones, the intensity measured in 10~ 6 dynes, <p measured from S. to E., N., W., and is measured above the horizon. The vector is the deflection from the mean position. Find the average for each zone. (Bigelow, pp. 324-325.) Time Arctic N Temperate Tropic S Temperate s e <P s <P s <p s <p a.m. 60 -36° 345° 15 -30° 111° 20 -33° 5° 19 27° 259° 1 63 -44 355 14 -35 109 19 -32 16 19 31 250 2 69 -43 5 14 -32 102 20 -36 7 17 35 251 3 74 -44 16 14 -33 108 20 -42 6 18 36 243 4 75 -42 25 15 -35 112 18 -34 10 20 36 226 5 77 -42 30 17 -33 110 17 -37 6 21 33 223 6 78 -40 32 20 -31 112 19 -36 4 24 31 222 7 76 -40 36 22 - 6 107 21 -37 339 26 24 235 8 65 -37 45 25 3 99 24 -30 297 28 28 248 9 54 -18 68 26 24 66 26 23 228 28 33 256 10 39 31 117 27 37 49 35 25 210 26 -27 296 11 47 44 195 25 38 312 43 22 204 25 -37 327 ADDITION OF VECTORS 61 4. Find the resultant attraction at a point due to a segment of a straight line which is (a) of uniform density, (6) of density which varies as the square of the distance from one end. What is the mean attrac- tion in each case? 5. Show that p = ta + \P$ is the equation of a parabola, that the equation of the tangent is p = Ua + \t\ 2 & + x(a + ttfi), that tangents from a given point are given by t = p ± V (p 2 — 2q), the point being pa + q/3, the chord of contact is p = — qP + y(a + PP) which has a direction independent of q so that all points of the line p = pa + zP have corresponding chords of contact which are parallel. If a chord is to pass through the point aa + bp for differing values of p, then q = ap — b and the moving point pa + qP lies on the line p — pa -f- (ap — b)P, whose direction is independent of b. 6. If a, /S, 7 are vectors to three collinear points, then we can find three numbers a, b, c such that aa + 6/S + cy = = a + b + c. 7. In problem 5 show that if three points are taken on the parabola corresponding to the values t\, U, tz, then the three points of intersection of the sides of the triangle they determine with the tangents at the vertices of the triangle are collinear. 8. Determine the points that divide the segment joining A and B, points with vectors a and 0, in the ratio I : m, both internally and ex- ternally. Apply the result to find the polar of a point with respect to a given triangle, that is, the line which passes through the three points that are harmonic on' the three sides respectively with the intersection of a line through the given point and the vertex opposite the side. 9. Show how to find the resultant field due to superimposed fields. 10. A curve on a surface is given by p = u(u, v), u = /(v), study the differential of p. CHAPTER V VECTORS IN A PLANE 1. Ratio of Two Vectors. We purpose in this chapter to make a more detailed study of vectors in a plane and the hypernumbers corresponding. In the plane it is convenient to take some assigned unit vector as a reference for all others in the plane, though this is not at all necessary in most problems. In fact we go back for a moment to the fundamental idea underlying the metric notion of number. According to this a number is defined to be the ratio be- tween two quantities of the same concrete kind, such as the ratio of a rod to a foot. If now we consider the ratio of vectors, regarding them as the same kind of quantity, it is clear that the ratio will involve more than merely numerical ratio of lengths. The ratio in this case is in fact what we have called a hypernumber. For every pair of vectors p, x there exists a ratio p : x and a reciprocal ratio x : p. This ratio we will designate by a roman character P p : x = p/x = IT That is to say, we may substitute p for qw. 2. Complex Numbers. If we draw p and x from one point, they will form a figure which has two segments for sides and an angle. (In case they coincide we still con- sider they have an angle, namely zero.) In this figure p is the initial side and x is the terminal side. Then their complex ratio is x : p. Since this ratio is to be looked upon as a multiplier, it is clear that if we were to reduce the sides in the same proportion, the ratio would not be changed. 62 VECTORS IN A PLANE 63 A change of angle would, however, give a different ratio. However, we will agree that all ratios are to be considered as equivalent, or as we shall usually say, equal, not only when the figures to which they correspond have sides in the same proportion, but also when they have the same angles and sides in proportion, even if not placed in the plane in the same position. For instance, if the vectors AB, AC make a triangle which is similar to the triangle DE, DF, if we take the sides in this order, then we shall consider that whatever complex or hypernumber multiplies AC into AB will also multiply DF into DE. This axiom of equivalence is not only important but it differentiates this particular hypernumber from others which might just as well be taken as fundamental. For instance, the Gibbs dyad of t : p is equally a hypernumber, but we cannot substitute for ir or p any other vectors than mere multiples of 7r or p. It is clear that in the Gibbs dyad we have a more restricted hypernumber than in the ordinary com- plex number which has been just defined, and which is a special case of the Hamiltonian quaternion. If we have a Gibbs dyad q, we can find the two vectors ir and p save as to their actual lengths. But with the complex number q we cannot find ir and p further than to say that for every vector there is another in the ratio q. In other words the only transformations allowed in the Gibbs dyad are transla- tion of the figure AB, AC or magnification of it. In the Hamiltonian quaternion, or complex number, the trans- formations of the figure AB, AC may be not only those just mentioned but rotation in the plane. In order to find a satisfactory form for the hypernumber q which we have characterized, we further notice that if we change the length of x in the ratio m then we must change q in the same ratio, and if we set for the ratio of the 64 VECTOR CALCULUS length or intensity of w to that of p the number r, it is evi- dent that we ought to take for q an expression of the form q = r<p(0), where <p(6) is a function of 0, the angle between p and t, only. Further if we notice that we now have 7T = r(p(6)p, where the first factor affects the change of length, the second the change of direction, it is plain that for a second multiplication by another complex number q' = r'<p(0'), we should have tt' = r , rcp(e , )i P {e) P = r'rip(W + 6)p. Whence we must consider that viO'Md) = <p{e f +$) = view). These expressions are functions of two ordinary numerical parameters, 0, 0', and are subject to partial differentiation, just like any other expressions. Differentiating first as to 0, then as to 6', we find (<p f being the derivative) <p\eM6') = ?'($+ e f ) = wmb), whence . v y) = V '{6') _ where & is a constant and does not depend upon the angle at all. It may, however, depend upon the plane in which the vectors lie, so that for different planes A; may be, and in fact is, different. N Since, when = the hypernumber becomes a mere numerical multiplier, <p'(0) = MO). If now we examine the particular function <p(0) = cos 0+k sin 6, VECTORS IN A PLANE 65 which gives <p'(d) = — * sin $ + k cos 6 = k cos 6 + k 2 sin 6, we find all conditions are satisfied if we take k 2 = — 1. We may then properly use this function to define <p. This very simple condition then enables us to define hyper- numbers of this kind, so that we write q = r(cos 6 + k sin 9) = r cks 6 = r g , where k 2 = — 1. 3. Imaginaries. It is desirable to notice carefully here that we must take k 2 equal to — 1, the same negative number that we have always been using. This is important because there are other points of view from which the character of k and k 2 would be differently regarded. For instance, in the original paper of Hamilton, On Algebraic Couples, the k, or its equivalent, is regarded as a linear substitution or operator, which converts the couple (a, b) into the couple (— b, a). While it is true that we may so regard the imaginary, it becomes at once obvious that we must then draw distinctions between 1 as an operator, and 1 as a number, and so for — 1, and indeed for any expression x + yi. In fact, such distinctions are drawn, and we find these operators occasionally called matrix unity, etc. From the point of view of the hypernumber, this distinction is not possible. Hypernumbers are extensions of the number system, similar to radicals and other algebraic numbers. The fact that, as we will see later, they are not in general commutative, does not prevent their being an extension. 4. Real, Imaginary, Tensor, Versor. In the complex number q = r cos 6 + r sin 6 • k the term r cos 6 is called the real part of q and may be written Rq. The term r sin 6-k is called the imaginary part of q 66 VECTOR CALCULUS and written Iq. The number r is called the tensor of q and written Tq. The expression cos 6 + sin • k is called the versor of </ and written Uq. Therefore, q= Rq+ Iq= TqUq. If q appears in the form q — a + bk we see at once that Rq = a , Iq= bk, Tq = V (a 2 + b 2 ), 6 = taiT^/a. 5. Division. If we have w = qp, then we also write p = g -1 7r. It becomes evident that &T l = RqKTqf, Iq-' = - Iql(Tq)\ Tq-* = 1/Tq, Uq- 1 = cos 6 — sin 6 • k. 6. Conjugate, Norm. The hypernumber q = Kq — Rq — Iq is called the conjugate of q. If q belongs to the figure AB, AC, then q belongs to an inversely similar triangle, that is, a similar triangle which has been reflected in some straight line of the plane. The product q° = Nq = (Tq) 2 is called the norm of q. It also has the name modulus of q, particularly in the theory of functions of complex variables. Evidently, Rq = i(q + q), Iq = h(q - ~q), r 1 = W* ^q~ l = Uq- 7. Products of Complex Numbers. From the definitions it is clear that the product of two complex numbers q, r, is a complex number s, such that Ts = TqTr,_ ZJ= zq+ Zr, Rqr = Rrq = Rqr = Rrq = RqRr - Tlqlr, Rqr = Rqr = Rrq = Rrq = RqRr + Tlqlr, Iqr = Irq = — Tqr = — Irq = Rqlr + Rrlq, Iqr = Irq = — Iqr = — Irq = Rrlq — Rqlr. Hence if Rqr = 0, the angles of q and r are complementary or have 270° for their sum. VECTORS IN A PLANE 67 If Rqr = 0, the angles differ by 90°. In particular we may take r = 1. If Iqr = 0, the angles are supplementary or opposite. If Iqr = 0, the angles are equal or differ by 180°. 8. Continued Products. We need only to notice that (qrs- • -z) = (z- • -srq). It is not really necessary to reverse the order here as the products are commutative, but in quaternions, of which these numbers are particular cases, the products are not usually commutative, and the order must be as here written. 9. Triangles. If ft y, 5, e are vectors in the plane, and e = gft 5 = gy f then the triangle of ft e is similar to that of y, 5, while if e = gft 5 = ?7, the triangles are inversely similar. These equations enable us to apply complex numbers to certain classes of problems with great success. 10. Use of Complex Numbers as Vectors. If a vector a is taken as unit, every vector in the plane may be written in the form qa, for some properly chosen q. We may therefore dispense with the writing of the a, and talk of the vector q, always with the implied reference to a certain unit a. This is the well-known method of Wessel, Argand, Gauss, and others. However, it should be noticed that we have no occasion to talk of q as a point in the plane. EXAMPLES (1) Calculate the path of the steam in a two-wheel tur- bine from the following data. The two wheels are rigidly connected and rotate with a speed a = 400 ° ft./sec. Be- 68 VECTOR CALCULUS tween them are stationary buckets which turn the exhaust steam from the buckets of the first wheel into those of the second wheel. The friction in each bucket reduces the speed by 12%. The steam issues from the expansion nozzle at a speed of /3 = 2200 2 o°. The proper exhaust angles of the buckets are 24°, 30°, 45°. Find the proper entrance angles of the buckets. 7 = relative velocity of steam at entrance to first wheel. = 2200 20 - 400o = 1830 24 .3. 8 = velocity of issuing steam, 88% of preceding, = 1610x56. € = entrance velocity to stationary bucket. = 5 + a = I6IO1M + 400o = 1255i4 8 .4. f = exit - 1105 30 . = entrance to next bucket = £ — a = 1105 30 — 400o = 78044.3. 77 = exit = 69O135. Absolute exit velocity = 690i35 + 400 = 495ioo. Steinmetz, Engineering Mathematics. (2). We may suppose the student is somewhat familiar with the usual elementary theory of the functions of a complex variable. If w is an analytic function of z, both complex numbers, then the real part of w, Rw, considered as a function of x, y or u, v, the two parameters which de- termine z, will give a system of curves in the x, y, or the u, v plane. These may be considered to be the transforma- tions of the curves Rw = const, which are straight lines parallel to the Y axis in the w plane. Similarly for the imaginary part. The two sets will be orthogonal to each other, since the slope of the first set will be ^— / -z — ; J * 1 1 dTIw/dTIw _ and 01 the other set ^ — / —^ — . But these are ox I dy VECTORS IN A PLANE 69 negative reciprocal, since dRw dTIw dRw_ dTIw ~ — n and ~ — ~ ox oy ay ox EXERCISES 1 . If a particle is moving with the velocity 12028° and enters a medium which has a velocity given by <r = P + 36 sin z [p, 0] 8 °, what will be its path? 2. The wind is blowing steadily from the northwest at a rate of 16 ft. /sec. A boat is carried round in circles with a velocity 12 ft. /sec. divided by the distance from the center. The two velocities are com- pounded, find the motion of the boat if it starts at the point p = 4 °. 3. A slow stream flows in at the point 12 ° and out at the point 12i8o°, the lines of flow being circles and the speed constant. A chip is floating on the stream and is blown by the wind with a velocity 640 . Find its path. 4. If a triangle is made with the sides q, r then R.qr is the power of the vertex with reference to the circle whose diameter is the opposite side. The area of the triangle is \TIqr. 5. The sum q + r can be found by drawing vectors qa, ra. 6. How is qr constructed? qr? 7. If OAE is a straight line and OCF another, and if EC and AF intersect in B, then OA BC + OC • AB + OB • CA = 0. If 0, A, B, C are concyclic this gives Ptolemy's theorem. 8. If ABC is a triangle and LM a segment, and if we construct LMP similar to ABC, LMQ similar to BCA, and LMR similar to CAB, then PQR is similar to CAB. 9. If the variable complex number u depends on the real number x as a variable parameter, by the linear fractional form ax + b u - ex + d then for different values of x the vector representing u will terminate on a circle. For if we construct b U ~d ' w = a u c 70 VECTOR CALCULUS this reduces to — (cx/d), hence the angle of w, which is the angle between u — ale and u — b/d, is the angle of — d/c and is therefore constant. Hence the circle goes through a/c (x = «) and b/d (x = 0). 10. If _ x(c — b)a + b(a — c) , U ~ k(c-b) + (a- c) where x is a variable real parameter, then the vector representative of u will terminate on the circle through A, B, C, where OA represents a, OB represents 6, and OC represents c. 11. Given three circles with centers C 1} d, C3, and O their radical center, P any point in the plane, then the differences of the powers of P with respect to the three pairs of circles are proportional respectively to the projections of the sides of the triangle CiC 2 Cz on OP. 12. Construct a polygon of n sides when there is given a set of points, Ci, C2, - • -, C n which divide the sides in given ratios a x : bi, a 2 : 62, • • •, a» : 6„. If the vertices are &, £ 2 , • • • , in, and the points Ci, C 2 , • • • , C n are at the ends of vectors 71, 72, •••, y n , we have Olll + &lfc = 7l(ai +6l) ' * * CLntn + bnh = 7n(a n + b n ). The solution of these equations will locate the vertices. When is the solution ambiguous or impossible? 13. Construct two directly similar triangles whose bases are given vectors in the plane, fixed in position, so that the two triangles have a common vertex. 14. Construct the common vertex of two inversely similar triangles whose bases are given. 15. Construct a triangle ABC when the lengths of the sides AB and AC are given and the length of the bisector AD. 1G. Construct a triangle XYZ directly similar to a given triangle PQR whose vertices shall be at given distances from a fixed point 0. Let the length of OX be a, of OY be 6, and of OZ be c. Then X is anywhere on the circle of radius a and center O. We have XY/XZ = PQIPR, that is, OY -OX = PQ OZ-OX PR' whence we have OXQR + OYRP + OZPQ = 0. We draw OXK directly similar to RPQ giving KO/OX = QR/RP and KO + OY + OZ -£§ = 0, that is, VECTORS IN A PLANE 71 In KOY we have the base KO and the length OY = b, and length of _ length PQ length RP' We can therefore construct KOY and the problem is solved. 17. The hydrographic problem. Find a point X from which the three sides of a given triangle ABC are seen under given angles. XB/XA = y cks 0, XC/XA = z cks p. XB = XA + AB, XC = XA + AC. Eliminate XA giving 2 cks <?•# A + y cksd-AC = BC. Find U such that z AJBI7 = Z AXC, Z ACtf = Z AXB, then BU = z cks *>. BA,CU = y cks OCA. Construct A ACX directly similar to A A UB. 18. Find the condition that the three lines perpendicular to the three vectors pa, qa, ra at their extremities be concurrent. We have p + xkp = q + ykq = r + zkr. Taking conjugates q — xkp = p — ykq = r— zkr. Eliminate x, y, z from the four equations. 19. If a ray at angle is reflected in a mirror at angle a the reflected ray is in the direction whose angle is 2 a — /3. Study a chain of mirrors. Show that the final direction is independent of some of the angles. 20. Show that if the normal to a line is a and a point P is distant y from the line, and from P as a source of light a ray is reflected from the line, its initial direction being — qa, then the reflected ray has for equation — 2ya + tqa = p. For further study along these lines, see Laisant: Theorie et Application des Equipollences. 11. Alternating Currents. We will notice an application of these hypernumbers to the theory of alternating currents and electromotive forces, due to C. P. Steinmetz. If an alternating current is given by the equation I = Io cos 2wf(t - h), the graph of the current in terms of t is a circle whose diameter is 7 making an angle with the position for t = of 2wfti. The angle is called the phase angle of the current. If two such currents of the same frequency are superim- 72 VECTOR CALCULUS posed on the same circuit, say we may set 7 = 7 cos 2irf(t - ti), F = Jo' cos 2tt/(* - fcO, sex 7 cos 2vfh + h' cos 2tt/V = 7 " cos 2wfh, 7 sin 2tt/<i + U sin 2u//i' = 7 " sin 2irft 2t 7" = 7 " cos 2wf(t - it), which also has for its graph a circle, whose diameter is the vector sum of the diameters of the other two circles. We may then fairly represent alternating currents of the simple type and of the same frequency by the vectors which are the diameters of the corresponding circles. The same may be said of the electromotive forces. If we represent the current and the electromotive force on the same diagram, the current would be indicated by a yellow vector (let us say) traveling around the origin, with its extremity on its circle, while at the same time the electromotive force would be represented by a blue vector traveling with the same angular speed around a circle with a diameter of different length perhaps. The yellow and the blue vectors would generally not coincide, but they would maintain an invariable angle, hence, if each is con- sidered to be represented by a vector, the ratio of these vectors would be such that its angle would be the same for all times. This angle is called the angle of lag, or lead, according as the E.M.F. is behind the current or ahead of it. The law connecting the vectors is E= ZI, where E is the electromotive force vector, that is, the vector diameter of its circle, 7 is the current vector, the diameter of its circle, and Z is a hypernumber called the impedance, VECTORS IN A PLANE 73 [<p/0], measured in ohms. The scalar part of Z is the resistance of the circuit, while the imaginary part is the reactance, the formula for Z being Z = r — xk. The value of x is 2irfL, where/ is the frequency, [T~ l ], and L is the inductance, [^G -1 ? 1 ], in henry s, or — l/2irfC where C is the permittance, [OT 1 ^ -1 ], in farads. [1 farad = 9- 10 11 e.s. units = 10 -9 e.m. units, and 1 ^nn/ = ^lO -11 e.s. units = 10 9 e.m. units.] It is to be noticed that reactance due to the capacity of the circuit is opposite in sign to that due to inductance. The law above is called the generalized Ohm's law. We may also generalize KirchofFs laws, the two generalizations being due to Steinmetz, and having the highest importance, inasmuch as by the use of these hypernumbers the same type of calculation may be used on alternating circuits as on direct circuits. The generalization of KirchofFs laws is as follows : (1) The vector sum of all electromotive forces acting in a closed circuit is zero, if resistance and reactance electro- motive forces are counted as counter electromotive forces. (2) The vector sum of all currents directed toward a distributing point is zero. (3) In a number of impedances in series the joint im- pedance is the vector sum of all the impedances, but in a parallel connected circuit the joint admittance (reciprocal of impedance) is the sum of the several admittances. The impedance gives the angle of lag or lead, as the angle of a hyper number of this type. We desire to emphasize the fact that in impedances we have physical cases of complex numbers. They involve complex numbers just as much as velocities involve positive 74 VECTOR CALCULUS of negative velocity, or rotations involve positive or nega- tive. We may also affirm that the complex currents and electromotive forces are real physical existences, every current implying a power current and a wattless current whose values lag 90° (as time) behind the power current. The power electromotive force is merely the real part of the complex electromotive force, and the wattless E.M.F. the imaginary part of the complex electromotive force, both being given by the complex current and the complex impedance. We find at the different points of a transmission line that the complex current and complex electromotive force satisfy the differential equations dl/ds = (g + Cok)E, dE/ds = (r + Look)L The letters stand for quantities as follows: g is mhos I mile, r is ohms/mile, C is farads/mile, L is henrys/mile. co = 2irf. Setting m* = (r + Lo>k)(g + Cirk), I 2 = (r + Lak)/(g + Cwk), so that m is [X -1 ] while / is ohms/mile, the solution of the equations is E = E cosh ms + ll sinh ms, I = Iq cosh ms + 1~ 1 Eq sinh ms, where E and 7 are the initial values, that is, where s = 0. If we set Eq = ZqIq and then set Z = Z cosh h, I = Z sinh h we have E = Z cosh (ms + h)I , I = l~ l Z sinh (ms + h)I , E = I coth (ms + h)I, E = sech h cosh (ms + h)E , I = csch h sinh (ms + h)I . To find where the wattless current of the initial station has become the power current we set I = kl , that is, sinh (ms -f- h) = k sinh h. VECTORS IN A PLANE 75 The value of s must be real. EXAMPLES (1) Let r = 2 ohms/mile, L = 0.02 henrys/mile, C = 0.0000005 farads/mile, g = 0, to = 2000, coL = 40 ohms/mile, conductor reactance, r + Look = 2 + 40/c ohms/mile impedance = 40.5 87 .i5 o . uC = 0.001 mhos/mile dielectric susceptance. g + Coik = 0.001 k mhos/mile dielectric admit- tance = 0.001 90 °. (g + Cuk)~ l = 1000/j" 1 = 1000 27 o° ohms/mile dielectric impedance. m 2 = 0.0405i 77 .i5°, m = 0.2001 88 .58°, P = 40500_.2.85°, I = 201.25_i.43°. Let the values at the receiver (s = 0) be E = 1000 o volts, 7 = o. Then we have E = 1000 cosh s0.2001 8 8.58°, for s = 100 E = 1000 cosh 20.01 88 . 58 = 625.9 45 .92°, I = 2.77 27 o, for s = 8 E = 50.01i26.ot, for s = 16 E = 1001i 80 .3°, for s = 15.7 E = 1000i 8 o°, a reversal of phase, for s = 7.85 E = 90 o. At points distant 31.4 miles the values are the same. If we assume that at the receiver end a current is to be maintained with Jo = 50 40 ° with E = 1000 o, E = 1000 cosh s0.2001 88 . 58 ° + 10062 38 . 5 7° sinh s0.2001 88 . 5 8°, I = 50 4 o° cosh sm + 5i. 4 3° sinh sm. At s = 100 E = 10730n355°. MacMahon, Hyperbolic Functions. 76 VECTOR CALCULUS (2) Let E - 10000, 7 - 65i 3 . 5 ° r = 1, g = 0.00002 Ceo = 0.00020 period 221.5 miles, o>L = 4. (3) The product P = EI represents the power of the alternating current, with the understanding that the fre- quency is doubled. The real or scalar part is the effective power, the imaginary part the wattless or reactive power. The value of TP is the total apparent power. The cos z P is the power factor, and sin / P is the induction factor. The torque, which is the product of the magnetic flux by the armature magnetomotive force times the sine of their angle is proportional to TIP, where E is the generated electromotive force, and/ is the secondary current. In fact, the torque is TI'EI-p/2irf where p is the number of poles (pairs) of the motor. 12. Divergence and Curl. In a general vector field the lines have relations to one another, besides having the peculiarities of the singularities of the field. The most important of these relations depend upon the way the lines approach one another, and the shape and position of a moving cross-section of a vector tube. There is also at each point of the field an intensity of the field as well as a direction, and this will change from point to point. Divergence of Plane Lines. If we examine the drawing of the field of a vector distribution in a plane, we may easily measure the rate of approach of neighboring lines. Starting from two points, one on each line, at the intersec- tion of the normal at a point of the first line and the second line, we follow the two lines measuring the distance apart on a normal from the first. The rate of increase of this normal distance divided by the normal distance and the distance traveled from the initial point is the divergence of the lines, or as we shall say briefly the geometric divergence of the field. It is easily seen that in this case of a plane VECTORS IN A PLANE 77 field it is merely the curvature of the curves orthogonal to the curves of the field. For instance, in the figure, the tangent to a curve of the field is a, the normal at the same point /5. The neighboring curve goes through C. The differential of the normal, which is the difference of BD and AC, divided by AC, or BD, is the rate of divergence of the second curve from the first for the distance AB. Hence, if we also divide by AB we will have the rate of angular turn of the tangent a in moving to the neigh- boring curve, the one from C. This rate of angular turn of the tangent of the field is the same as the rate of turn of the normal of the orthogonal system, and is thus the curva- ture of the normal system. Curl of Plane Lines. If we find the curvature of the original lines of the field we have a quantity of much im- portance, which may be called the geometric curl. This must be taken plus when the normal to the field on the convex side of the curve makes a positive right angle with the tangent, and negative when it makes a negative right angle with the tangent. Curl is really a vector, but for the case of a plane field the direction would be perpendicular to the plane for the curl at every point, and we may con- sider only its intensity. Divergence of Field. Since the field has an intensity as well as a direction, let the vector characterizing the field be cr = Ta-a. Then the rate of change of TV in the direc- tion of a, the tangent, is represented by d a T<r. Let us now consider an elementary area between two neighboring curves of the field, and two neighboring normals. If we consider Ta as an intensity of some quantity whose amount 78 VECTOR CALCULUS depends also upon the length of the infinitesimal normal curve, so that we consider the quantity Ta-dn, then the value of this quantity, which we will call the transport of the differential tube (strip in the case of a plane field), TV being the density of transport, will vary for different cross-sections of the tube, and for the case under considera- tion, would be Ta'dn' - Tadn. But TV' = TV + d a Ta-ds and dn' = dn + ds-dn times the divergence of the lines. Therefore, the differential of the transport will P" ~T~ ( be (to terms of the first order) ds X dn X ( TV I L— times divergence + d a Ta). Hence, the density F ' of this rate of change of the transport is TV . times the divergence + the rate of change of TV along the tangent of the vector line of the field. This quan- tity we call the divergence of the field at the initial point, and sometimes it will be indicated by div. cr, sometimes by — SVa, a notation which will be explained. It is clear that if the lines of a field are perpendicular to a set of straight lines, since the curvature of the straight lines is zero, the divergence of the original lines is zero, and the expression reduces to d a T<r. Curl of Field. We may also study the circulation of the vector a along its lines, by which we mean the product of the intensity TV by a differential arc, that is, Tads. On the neighboring vector line there is a different intensity, TV', and a different differential arc ds'. The differential of the circulation is easily found in the same manner as the divergence, and turns out to be — (dfiTa + TV X curl of the vector lines). This quantity we shall call the curl of the field, written sometimes curl a, and more frequently Wa, which notation will be explained. 1 VECTORS IN A PLANE 79 It is evident that the curl of a is the line integral of the Tads around the elementary area, for the parts contributed by the boundary normal to the field will be zero. Hence, we may say that curl a is the limit of the circulation of <r around an elementary area constructed as above, to the area enclosed. We will see later that the shape of the area is not material. Likewise, the divergence is clearly the ratio to the elemen- tary area of the line integral of the normal component of <r along the path of integration. We will see that this also is independent of the shape of the area. Further, we see that in a field in which the intensity of a is constant the divergence becomes the geometric divergence times the intensity TV, and the curl becomes the geometric curl times the intensity T<r. Divergence and curl have many applications in vector analysis in its applications to geometry and physics. These appear particularly in the applications to space. A simple example of convergence or divergence is shown in the changing density of a gas moving over a plane. A simple caSfc of curl is shown by a needle imbedded in a moving viscous fluid. The angular rate of turn of the direction of the needle is one-half the curl of the velocity. 13. Lines as Levels. If the general equation of a given set of curves is u(x, y) = c, the§e curves will be the vector lines of an infinity of fields, for if the differential equation of the lines is dx/X m dy/Y, then we must have Xdu/dx + Ydu/dy = and for the field a = Xa + Y0. 80 VECTOR CALCULUS We may evidently choose X arbitrarily and then find Y uniquely from the equation. However, if a\ is any one field so determined, any other field is of the form a = <TiR(x, y). The- orthogonal set of curves would have for their finite equation v(x, y) = c and for their differential equation Xdvldy - Ydv/dx = 0. If we use a uniformly to represent the unit tangent of the u set, and P the unit tangent of the v set, then P = ha. The gradient of the function u is then d u-(3, and the gradient of the function v is — d a v-a. But the gradient of u is also (u x , u y ) and of v is (v x , v v ) = (u Vf — u x ). It follows that the tensors of the gradients are equal. In fact, writing Vm for gradient u, we have Vt> = kVu. We also have for whatever fields belong to the two sets of orthog- onal lines for u curves, a = rVv, for the v curves, a' = sVu, or also we may write Vv = tot, Vu = tp, a = Ta-ct. 14. Nabla. The symbol V is called nabla, and evidently may be written in the form ad/dx + Pd/dy for vectors in a plane. We will see later that for vectors in space it may be written ad/dx + Pd/dy + yd/dz, where a, ft y are the usual unit vectors of three mutually perpendicular directions. However, this form of this very important differential operator is not at all a necessary form. In fact, if a and fi are any two perpendicular unit vectors in a plane, and dr, ds are the corresponding differential dis- tances in these two directions, then we have V = ad/dr + pd/ds. VECTORS IN A PLANE 81 For instance, if functions are given in terms of r, 6, the usual polar coordinates, then V = Upd/dr + kUpd/rdd. The proof that for any orthogonal set of curves a similar form is possible, is left to the student. In general, V is defined as follows : V is a linear differentiating vector operator connected with the variable vector p as follows: Consider first, a scalar function of p, say F(p). Differentiate this by giving p any arbitrary differential dp. The result is linear in dp, and may be looked upon as the product of the length of dp and the projection upon the direction of dp of a certain vector for each direction dp. If now these vectors so projected can be reduced to a single vector, this is by definition VF. For instance, if F is the distance from the origin, then the differential of F in any direction is the projection of dr in a radial direction upon the direc- tion of differentiation. Hence, V7p = Up. In the case of plane vectors, VF will lie in the plane. In case the differential of F is polydromic, we define VF as a poly- dromic vector, which amounts to saying that a given set of vectors will each furnish its own differential value of dF. In some particular regions, or at certain points, the value of J7F may become indefinite in direction because the differentials in all directions vanish. Of course, functions can be defined which would require careful investigation as to their differentiability, but we shall not be concerned with such in this work, and for their adequate treatment reference is made to the standard works on analysis. We must consider next the meaning of V as applied to vectors. It is evident that if V is to be a linear and there- fore distributive operator, then such an expression as Va must have the same meaning as VXa. + V Y(3 + VZy if a = Xa + F/3 -r Zy, where a, 0, y are any independent constant vectors. This serves then as the definition of 82 VECTOR CALCULUS Vo-, the only remaining necessary part of the definition is the vector part which defines the product of two vectors. This will be considered as we proceed. 15. Nabla as a Complex Number. We will consider now p to represent the complex number x + yk, or r e , and that all our expressions are complex numbers. The proper expression for V becomes then V = d/dx + kd/dy = Upd/dr + kUpd/rdd. In general for the plane, let p depend upon two parameters u, v, and let dp = p\du -f- p 2 dv. If a is a function of p (generally not analytic in the usual sense) and thus dependent on u, v, we will have da = dcr/du-du + da/dv-dv = R-dpV -a. If we multiply dp by kpi, which is perpendicular to pi, the real part of both sides will be equal and we have, since kpi is perpendicular to pi, Rkpidp — dvRkpip 2 , and similarly Rkpidp = duRkpipi = — duRkpip 2 since the imaginary part of pip 2 equals — the imaginary part of p 2 pi- Substituting in da we have A, = «.*,(-,*£- £+#-£) <r. \ Rkpip 2 ou Rkpip 2 dvJ The expression in (), however, is exactly what we have de- fined above as V, and thus we have proved that we may write V in the form corresponding to dp in terms of u and v : V = k(p 2 d/du — pid/dv)/Rkpip 2 . In case pi and p 2 are perpendicular the divisor evidently VECTORS IN A PLANE 83 reduces to ± Tp\Tp 2 according as p 2 is negatively perpendic- ular to pi or positively perpendicular to it. We may write V in this case in the form (since p 2 = — kpi- Tp 2 /Tpi or + kpr Tp 2 /Tpi) v = _pi_A . _Pi_ A = p f ii, p -i 1_ . T Pl 2 du^ T P2 2 dv F du^ Ht dv In any case we have dF = Rdp\/F, da = Rdp\7 -v. Also in any case V = Vu-d/du + \7v-d/dv. 16. Curl, Divergence, and Nabla. Suppose now that a is the complex number for the unit tangent of one of a set of vector lines, and |8 the complex number for the unit tangent of the orthogonal set, at the same point. The curvature of the orthogonal set is the intensity of the vector rate of change of (3 along the orthogonal curve. But this is the same as the rate of change of the unit tangent a as we pass along the orthogonal curve from one vector line to an adjacent one. The differential of a is perpendicular to a, and hence parallel to the direction of /3. Hence this curvature can be written But if we also consider the value of R- a(R-aV)a, since the differential of a in the direction of a has no component parallel to a, this term is zero, and may be added to the preceding without affecting its value. Hence the curvature of the orthogonal set reduces to R(aRaV + ^/3V)« = R-Va. This is the divergence of the curves of a. If now <j = Tcr-a, we find from the definition of the divergence of a that it is merely R-Va. Considering in the same manner the definition of curl of a, 84 VECTOR CALCULUS we find it reduces to — R-kV<r, and if we multiply this by k, so that we have curl a = - kRW(T=LV<r f we see at once that when added to the expression for the divergence of a we have div-<7 + curl <r = V<r. The real part of this expression is therefore the divergence of a, and the imaginary part is the curl of a. This will agree with expressions for curl and divergence for space of three dimensions. We have thus found some of the remarkable properties of the operator V . 17. Solenoidal and Lamellar Vector Fields. When the divergence of a is everywhere zero, the field is said to be solenoidal. If the curl is everywhere zero, the field is called lamellar. 18. Properties of the Field. Let a set of curves u = c be considered, and the orthogonal set v — a, and let the field a be expressed in the form o- = XVu + FVfl, where it is assumed that the gradients Vu, Vv exist at all points to be considered. We have then diver = RVa = RvXVu+ RvYVv _ + XRWu+ YRWv. The expression RWu is called the plane dissipation of u. In case it vanishes it is evident that u satisfies Laplace's equation, and is therefore harmonic. We also have curl o- = I- V<r = — kRkVXVu — kRkvYVv, the other parts vanishing. VECTORS IN A PLANE 85 Since we have chosen orthogonal sets of curves we may write these in the forms diver = (TVu) 2 dX/du + (TVv) 2 dY/dv + XRVVu + YRvVv, curl o- = (TVu)(TW)(dY/du - dX/dv)k. In case we have chosen the lines of cr for the u curves, then X = 0, and a = Y V v diver = YRVW+ (TVv) 2 dY/dv, curl (7= TVuTVvdY/du-k. We notice that curl Vu = 0, curl Vv = 0, div k\/u — 0, divkVv = 0, kVu = VvTVu/TVv, and for Y = TVu/TVv, we have (TVu)- 2 RvS7u = d log (TVu/TVv)/du, ' (fV*)~*BVV« = d log (TVv/TVu)/dv. We may now draw some conclusions as to the types of curves and <r. (Cf. B. O. Peirce, Proc. Amer. Acad. Arts and Sci., 38 (1903) 663-678; 39 (1903) 295-304.) (1) The field will be solenoidal if diver = 0, hence d log Y/dv = - RVW/TW 2 , ' which may be integrated, giving Y = e f(u ' v) + o{u) . If v is harmonic, Y is a function of u only and a =G(u)Vv. (2) If the field is lamellar, curl a = 0, and Y is a function of v only, so that a = H(v)Vv = VL(v). (3) If the field is both solenoidal and lamellar, RVVL(v) = 0, whence RVVv/(TVv) 2 = /(*), which is a condition on the character of the curves. Hence 86 VECTOR CALCULUS it is not possible to have a solenoidal and lamellar field with purely arbitrary curves. (4) If the field is solenoidal and Ta, the intensity, is a function of u alone, Y = p(u)/TVv, and therefore d log Y/dv = - dTVv/TVvdv = - RvVv/TVv 2 , whence 2RVW = d(TVv) 2 /dv, which is a condition on the curves. An example is the cross-section of a field of magnetic intensity inside an in- finitely long cylinder of revolution which carries lengthwise a steady current of electricity of uniform current density. (5) If a is lamellar and Ta is a function of v only, TVv = g(v). An example is the field of attraction within a homogeneous, infinitely long cylinder of revolution. The condition is a restriction on the possible curves. (6) If the field is lamellar and Ta a function of u only, since Y is a function of v only, d log TVv/du = k(u), or TVv = l(u)/m(v). This restricts the curves. (7) If the field is solenoidal and Ta a function of v only, Ta = p(v)TW. Therefore d log Ta/dv = d log TVa/dv — RS7Vv/(T\7v) 2 . Hence either both sides are constant or else both expressible in terms of v. If the field is not lamellar also, TVv must then be a function of u as well as of v. (8) If the field is lamellar and has a scalar potential function, that is, a = VP, then since a = q(v)Vv, we must have P a function of v only, and a = P'Vfl. From this it follows that diver = P\v)RVVv + P"(v)(TVv) 2 . (9) If the field is uniform, Ta — a, Y = a/T\7v, and a is lamellar only if TVv is either constant or a function of v only, while a is solenoidal only if we have 2RVW = d(TVv) 2 /dv. VECTORS IN A PLANE 87 (10) Whatever function u is, the u lines are vector lines for the vectors £ = f(u)UVv, f = g(v)U\7v, or T? = *(«, r)tTVf. (11) If the field is solenoidal, TV a function of u only, and the w curves are the lines of the field, then the curl takes the form — k div • ka, whence it has the form k[b(u)RVVu+ b'(u)(TVu) 2 ], where b may be any differentiate function. If TV is also a function of v, the form of the curl is k[b(u, v)RVVu + db(u, v)/du(TVu) 2 ]. (12) If TV is a function of u only, the divergence takes the form diver - Ta[RWv/TVv - dTVv/dv]. (13) If TV is a function of v only curl a = - kTaTVu/TVv-dTVv/du. 19. Continuous Media. When the field is that of the velocity of a continuous medium, we have two cases to take into account. If the medium is incompressible it is called a liquid, otherwise a gas. Incompressibility means that the density at a point remains invariable, and if this is c, then from dc/dt= dc/dt + RaVc, = dc/dt + RV(ca) - cRV<r we see that the first two terms together vanish, giving the equation of continuity, since they give the rate per square centimeter at which actual material (density times area, since the height is constant) is changing. Hence in this case dc/dt = — cRV<t> This gives the rate of change of the density at a point moving with the fluid. Hence if it is incompressible, the velocity is solenoidal, RV& = 0. 88 VECTOR CALCULUS This may also be written curl (— ka) = 0, hence — ka = V?, and <j — kvQ, which shows that for every liquid there is a function Q called the function of flow. When curl { = 0, we have seen that £ is called lamellar. It may also be called irrotational, since the curl is twice the angular rate of rotation of the infinitesimal parts of the medium, about axes perpendicular to the plane, and if curl { = there is no such rotation. Curl is analogous to density, being a density of rotation when the vector field is a velocity field. The circulation of the field is the integral fRadp along any path from a point A to a point B. This is the same as Xdx + Ydy, and is exact when dX/dy = dY/dx. But this gives exactly the condition that the curl should vanish. Hence if the motion is irrotational the circulation from one point to another is independent of the path. In this case we may write a = VP where P is called the velocity potential. When a is irrotational, the lines of Q have as orthogonals the lines of P. If the motion is rotational, these orthogonals are not the lines of such a function as P. If the motion is irrotational, we have for a liquid, RwP = 0, and P must be harmonic. Hence if the orthogonal curves of the Q curves can belong to a harmonic function they can be curves of a velocity potential. If a set of curves belong to the harmonic function u, then RWu = 0, and this shows that the curl of — JcVu is zero, whence Rdp(— k\/u) is exact = dv, where Vv = — kVu. From this we have Vm = kvQ for the condition that the orthogonal curves belong to a harmonic function. This however gives the equation TS/u = TvQ. We may assert then for a liquid that there is always a function of flow, and the curves belonging to VECTORS IN A PLANE 89 this function are the vector lines of the velocity, the in- tensity of the velocity being the intensity of the gradient of the function of flow. If the orthogonal curves belong to a function which has a gradient of the same intensity, both functions are harmonic, the function of the orthogonal set is a velocity potential, and the motion is irrotational. We have a simple means of discovering the sets of curves that belong to harmonic functions, as is well known to students of the theory of functions of a complex variable, since the real and the imaginary part of an analytic function of a complex variable are harmonic for the variable co- ordinates of the variable. That is to say, if p = x + yk, and £ = /(p) = u -\- vk, then u, v are harmonic for x, y. The condition given by Cauchy amounts to the equation Vm = — k\/v, or V£ = where £ is a complex number. It is clear from this that the field of £ is both solenoidal and lamellar, a necessary and sufficient condition that £ be an analytic function of a complex variable. In this case £ is called a monogenic function of position in the plane. It is clear that £ = VH where H is a harmonic function. In case there are singularities in the field it is necessary to determine their effect on the integrals. For instance, if we have a field a and select a path in it, from A to B, or a loop, the flux of a through the path is the integral of the projection of a on the normal of the path, that is, if the path is a curve given by dp, so that the projection is Ra(— kdp), the integral of this is the flux through, the path. It is written 2 = SI (- Rakdp) = - kfladp. In the case of a liquid the condition RV<r = shows that the expression is integrable over any path from A to B, with the same value, unless the two paths enclose a singu- larity of the field. In the case of a node, the integral around 7 90 VECTOR CALCULUS a loop enclosing the node is called the strength of the source or sink at the node. We may imagine a constant supply of the liquid to enter the plane or to leave it at the node, and be moving along the lines of the field. Such a system was called by Clifford a squirt If the circulation is taken around a singular point it will usually have a different value for every turn around the point, giving a polydromic function. These peculiarities must be studied carefully in each case. EXERCISES 1. From £ = Ap n we find in polar coordinates that u = Ar n cos nd, v = Ar n sin nd. These functions are harmonic and their curves orthogonal. Hence if we set a = Vwora = V#, we shall have as the vector lines of <r the v curves or the u curves. What are the curves for the cases n = — 3, — 2, — 1, 1, 2, 3? What are the singularities? 2. Study £ = A log p, and £ = A log (p — a)/(p + a). 3. Consider the function given implicitly by p = £ + e*. This represents the flow of a liquid into or out of a narrow channel, in the sense that it gives the lines of flow when it is not rotational. 4. Show that a = A/p gives a radial irrotational flow, while a = Ak/p gives a circular irrotational flow. What is true of a = Akpl The last is Clifford's Whirl. 5. Study a flow from a source at a given point of constant strength to a sink at another point, of the same strength as the source. 6. If the lines are concentric circles, and the angular velocity of any particle about the center is proportional to the n-th power of the radius of the path of the point, show that the curl is \ {n + 2) times the angular velocity. 7. A point in a gas is surrounded by a small loop. Show that the average tangential velocity on the loop has a ratio to the average normal velocity which is the ratio of the tensor of the curl to the divergence. 8. What is the velocity when there is a source at a fixed origin, and the divergence varies inversely as the w-th power of the distance from the origin. [The velocity potential is A log r — B{n — 2)~ 2 r 2-n .] 9. Consider the field of two sources of equal strength. The lines are for irrotational motion, cassinian ovals, where, if r, r' are the distances VECTORS IN A PLANE 91 from the two sources (foci) and rr' — h 2 , Q = A log h + B, the velocity is such that T<r = ATp/h 2 , the origin being half way between the foci; the orthogonal curves are given by u = iA[ir/2 — (0 + di)] where 0, 0i are the angles between the axis and the radii from the foci, that is they are equilateral hyperbolas through the foci. The circulation about one focus is ttA, about both 2irA. 10. If the lines are confocal ellipses given by z 2 /m + i/VG* - c 2 ) = 1, then Q = A log ( \V + V (m — c 2 )) + B. If p is the perpendicular from the center upon the tangent of the ellipse at any point, then the velocity at the point is such that T<r = — Ap/ y/ [/*(/* — c 2 )], and the direction of <r is the unit normal. The potential function is A sin -1 B' V v\c. V v is the semi-major axis. What happens at the foci? 11. If the stream lines are the hyperbolas of the preceding, then a = 2 A V (*7(m — v)) times the unit normal of the hyperbola. On the line p = yka there is no velocity, at the foci the velocity is oo , half way between it is 0. The lines along the major axis outside the foci act like walls. 12. If we write for brevity u x for T\7u, and vi for T\7v, show that we have whether the u curves are orthogonal to the v curves or not, V V = Ui 2 d 2 jdu 2 + Vi 2 d 2 ldv 2 + VVud/du + VVvd/dv + 2RVuVvd 2 /dudv. If the sets of curves are orthogonal the last term vanishes; if u and v are harmonic the third and fourth terms drop out; if both cases happen, only the first two terms are left. 13. In case of polar coordinates, Vr = Up, V0 = r" 2 A;p and VV = d 2 /dr 2 + r~ l dldr + r _2 d 2 /d0 2 . 14. A gas moves in a plane in lines radiating from the origin, which is a source. The divergence is a function of r only, the distance from the center. Find the velocity and the density at any point. a = p f{r), flVo- = eir) = 2/(r) + rf'{r), and f(r) = Ajr 2 + r~ 2 fre{r)dr. To determine c, RV log co- = - e(r) = f(r)Rp\/ log c = rf(r)d log c/dr. 15. Show that in the steady flow of a gas we may find an integrating factor for Rdpka by using the density, [dc/dt = = Rsjca = curl -fcco-, and Rdpkca is exact.] 16. A fluid is in steady motion, the lines being concentric circles. The curl is known at each point and the tensor of a is a function of r only. Find the velocity and the divergence. 92 VECTOR CALCULUS 17. Rotational motion, that is a field which is not lamellar, is also called vortical motion. The points at which the curl does not vanish may be distributed in a continuous or a discontinuous manner. In fact there may be only a finite number of them, called vortices. We have the following: <r = k\7Q, VVQ = T curl a = 2«, Q = 7r _1 //«' log rdx'dy' + Q , where «' denotes co at the variable point of the integration, r is the variable distance from the point at which the velocity is wanted, and Q is any solution of Laplace's equation which satisfies the boundary conditions. If the mass is unlimited and is stationary at infinity we have « = kfwfftt'ifi - P ')/T(p - py-dx'dy'. A single vortex filament at p of strength I would give the velocity a =U2T.(p-p')IT(p-p')\ If we multiply the velocity at each point p at which there is a vortex by the strength, and integrate over the whole field, we find the sum is zero. There is then a center of vortices where the velocity is zero, something like a center of gravity. Instances are (1) A single vortex of strength I. The vortex point will remain at rest, and points distant from it r will move on concentric circles with the vortex as center, and velocity l/2wr. The circulation of any loop surrounding the vortex is of course the strength. (2) Two vortices of strengths k, U. They will rotate about the common center of gravity of two weighted points at the fixed distance apart a, the weights being the two strengths. The angular velocity of each is 27ra 2 The stream lines of the field are given by fxhf % h = const. When k = — {, the center is at infinity, and the vortices remain a fixed dis- tance apart, moving parallel to the perpendicular bisector of this segment joining them. Such a combination is called a vortex pair. The stream lines of the accompanying velocity are coaxal circles referred to the moving points as limit points. The plane of symmetry may be taken as a boundary since it is one of the stream lines, giving the motion of a single vortex in a field bounded by a plane, the linear velocity of the vortex being parallel to the wall and \ of the velocity of the liquid along the wall. The figure suggests the method of images which can indeed be applied. For further problems of the same character works on Hydrodynamics should be consulted. VECTORS IN A PLANE 93 18. Liquid flows over an infinite plane towards a circular spot where it leaks out at the rate of 2 cc. per second for each cm. 2 area of the leaky portion. The liquid has a uniform depth of 10 cm. over the entire plane field. Find formulas for the velocity of the liquid inside the region of the leaky spot, and the region outside, and show that there is a potential in both regions. a = iVp in spot, 40/p outside, P = ^pp in spot, 40 log Tp — 20 log 400 outside. Find the flux through a plane area 20 cm. long and 10 cm. high, whose middle line is 5 cm. from the center of the leaky spot, also when it is 30 cm. from the leaky spot. Find the divergence in the two regions. Franklin, Electric Waves, pp. 307-8. 19. Show that in an irrotational motion with sources and sinks, the lines of flow are the orthogonal curves of the stream lines of a correspond- ing field in which the sources and the sinks are replaced by vortices of strengths the same as that of the sources and sinks, and inversely. Stream lines and levels change place as to their roles. For sources and sinks Q = 1/2tt-ZZi0i, P = 1/2* -Z log r x h. 20. Vector Potential. In the expression a = — VkQwe express (rasa vector derived by the operation of V upon — JcQ, the latter being a complex number. In such a case we may extend our terminology and call — JcQ the vector potential of a. A vector may be derived from more than one vector potential. In order that there be a vector potential it is necessary and sufficient that the divergence of <t vanish. Hence any liquid flow can have a vector potential, which is indeed the current function multiplied by — k. It is clear that Q must be harmonic. CHAPTER VI VECTORS IN SPACE 1. Biradials. We have seen that in a plane the figure made up of two directed segments from a vertex enables us to define the ratio of the two vectors which constitute the sides when the figure is in some definite position. This ratio is common to all the figures produced by rotating the figure about a normal of the plane through its vertex, and translating it anywhere in the plane. We may also reduce the sides proportionately and still have the same ratio. The ratio is a complex number or, as we will say in general, a hypernumber. If now we consider vectors in space of three dimensions, we may define in precisely the same manner a set of hyper- numbers which are the ratios of the figures we can produce in an analogous manner. Such figures will be called biradials. To each biradial there will correspond a hyper- number. Besides the translation and the rotation in the plane of the two sides of the biradial, we shall also permit the figure to be transferred to any parallel plane. This amounts to saying that we may choose a fixed origin, and whatever vectors we consider in space, we may draw from the origin two vectors parallel and equal to the two con- sidered, thus forming a biradial with the origin as vertex. Then any such biradial will determine a single hyper- number. Further the hypernumbers which belong to the biradials which can be produced from the given biradial by rotating it in its plane about the vertex will be con- sidered as equal. 94 VECTORS IN SPACE 95 The hypern umbers thus defined are extensions of those we have been using in the preceding chapter, the new feature being the different hypernumbers k which we now need, one new k in fact for each different plane through the given vertex. This gives us then a double infinity of hypernumbers of the complex type, r-cks 6, where the double infinity of k's constitute the new elements. 2. Quaternions. The hypernumbers we have thus de- fined metrico-geometrically involve four essential param- eters in whatever way they are expressed, since the biradials involve two and the plane in which they lie two more. Hence they were named by Hamilton Quaternions. In order to arrive at a fuller understanding of their prop- erties and relations, we will study the geometric properties of biradials. In the first place if we consider any given biradial, there is involved in its quaternion, just as for the complex number in the preceding chapter, two parts, a real part and an imaginary part, and we can write the quaternion in the form q = r cos 6 + r sin 6 -a, where a corresponds to what was written k in the preceding chapter, and is a hypernumber determined solely by the plane of the biradial. On account of this we may properly represent a by a unit normal to the plane of the biradial, so taken that if the angle of the biradial is considered to be positive, the direction of the normal is such that a right- handed screw motion turning the initial vector of the biradial into the terminal vector in direction would in- volve an advance along the normal in the direction in which it points. It is to be understood very clearly that the unit vector a and the hypernumber a are distinct entities, one merely representing the other. The real 96 VECTOR CALCULUS part of q is called, according to Hamilton's terminology, the scalar part of q, and written Sq. The imaginary part is called, on account of the representation of a as a vector, the vector part of q and written Vq. The unit a is called the unit vector of q and written UVq. The angle of q is and written Zq. The number r which is the ratio of the lengths of the sides of the biradial is called the tensor of q, and written Tq. The expression cos 6 + sin 6 -a = cas-d is called the versor of q, and written Z7^. Sq is a quaternion for which = 0° or 180°, Fg is a quaternion for which = 90° or 270°. Tq is a quaternion of 0°, being always positive, a is a quaternion of = 90°, and sometimes called a right versor. 3. Sum of Quaternions. In order to define the sum of two quaternions we define the sum of two biradials first. This is accomplished by rotating the two biradials in their planes until their initial lines coincide, and then diminishing or magnifying the sides of one until the initial vectors are exactly equal and coincide. This is always possible. We then define as the sum of the two biradials, the biradial whose initial vector is the common vector of the two, and terminal vector is the vector sum of the two terminal vectors. The sum of the corresponding quaternions is then the quaternion of the biradial sum. Since vector addition is commutative, the addition of quaternions is commutative. Passing now to the scalar and vector parts of the quater- nions, we will prove that they can be added separately, the scalar parts like any numbers and the vector parts like vectors. In the figure let the biradial of q be OB/OA, of r be OC/OA, and of q + r be OD/OA. Let the vector part of q, Tq- sin Zq-UVq be laid off as a vector Vq perpendicular VECTORS IN SPACE 97 to the plane of the biradial of q, and similarly for Vr. Then we are to show that V(q + r) = Vq + Vr in the representation and that this represents the vector part of q + r according to the definition. It is evident that OB = OB' + B'B, the first vector along OA, the second perpendicular to OA. Also OC = OC" + C"& + C'C, the first part along OA, the second parallel to B'B, and the third perpendicular to the plane of OAB. The sum OB + OC = OD, where OD = OB" + D"D' + D'Z), and 0Z>" - 05' + 00", D"£' = B'B + C ,, (7 , , D'D = C'C. Hence the biradial of the sum is OD/OA, where the scalar part is the ratio of OD" to OA. This is clearly the sum of the scalar parts of q and r, and S( q + f ) = Sq+ Sr. The vector part of the quaternion for OD/OA is the ratio of D"D to OA in magnitude, and the unit part is repre- sented by a unit normal perpendicular to OD" and D"D. But D"D = B'B + C'C, and the ratio of D"D to OA equals the sum of the ratios of B'B and C'C to OA. If then we draw, in a plane through which is perpendicular to OA, the vector Vq along the representative unit normal of the plane OAB, and of a length to represent the numerical ratio of B'B to OA, and likewise Vr to represent the ratio of C'C to OA laid off along the representative unit normal 98 VECTOR CALCULUS to the plane OAC, because D"D is parallel to this plane, as well as B'B and C"C, the representative unit vector of q+ r will lie in the plane, and will be in length the vector sum of Vq and Vr, that is V(q + r) as shown. It follows at once since the addition of scalars is associa- tive, and the addition of vectors is associative, and the two parts of a quaternion have no necessary precedence, that the addition of quaternions is associative. 4. Product of Quaternions. To define the product of quaternions we likewise utilize the biradials. In this case however we bring the initial vector of the multiplier to coincide with the terminal line of the multiplicand, and define the product biradial as the biradial whose initial vector is the initial vector of the multiplicand, and the terminal vector is the terminal vector of the multiplier. In the figure, the product of the biradials OB/OA, and Fig. 13. OC/OB, is, writing the multiplier first, OC/OB- OB/OA = OC/OA. It is clear that the tensor of the product is the product of the tensors, so that T-qr= TqTr. It follows that U-qr = UqUr. It is evident from the figure that the angle of the product will be the face angle of the trihedral, AOC, or on a unit sphere would be represented by the side of the spherical VECTORS IN SPACE 99 triangle corresponding. It is clear too that the reversal of the order of the multiplication will change the plane of the product biradial, usually, and therefore will give a quaternion with a different unit vector, though all the other numbers dependent upon the product will remain the same. However we can prove that multiplication of quaternions is associative. In this proof we may leave out the tensors and handle only the versors. The proof is due to Hamilton. To represent the biradials, since the vectors are all taken as unit vectors, we draw only an arc on the unit sphere, from one point to the other, of the two ends of the two unit vectors of the biradial. Thus we represent the biradial of q by CA, or, since the biradial may be rotated in its plane about the vertex, equally by ED. The others in- volved are shown. The product qr is represented by FD, from the definition, or equally by LM. What we have to prove is that the product p • qr is the same as the product pq-r, that is, we must prove that the arcs KG and LN are on the same great circle and of equal length and direction. Fig. 14. Since FE = KH, ED - CA, HG = CB, LM = FD, the points L, C, G, D are on a spherical conic, whose cyclic planes are those of AB, FE, and hence KG passes through L, and with LM intercepts on AB an arc equal to AB. That is, it passes through N, or KG and LN are arcs of the 100 VECTOR CALCULUS same great circle, and they are equal, for G and L are points in the spherical conic. 5. Trirectangular Biradials. A particular pair of bira- dials which lead to an interesting product is a pair of which the vectors of each biradial are perpendicular unit vectors, and the initial vector of one is the terminal of the other, for in such case, the product is a biradial of the same kind. In fact the three lines of the three biradials form a tri- rectangular trihedral. If the quaternions of the three o Fig. 15. are i, j, k, then we see easily that the quaternion of the biradial OC/OB is represented completely by the unit vector marked i, the quaternion of OA/OC by j, and of OB/OA by k. The products are very interesting, for we have ij = k, jk = i, hi = j, and if we place the equal biradials in the figure we also have ji = — k, kj = — i, ik = — j. Furthermore, we also can see easily that, utilizing the common notation of powers, V- = - 1, ? - - 1, V - ■■- 1. Since it is evidently possible to resolve the vector part of any quaternion, when it is laid off on the unit vector of its plane as a length, into three components along the direc- tions of i, j, k, and since the sum of the vector parts of VECTORS IN SPACE 10] quaternions has been shown to be the vector part of the sum, it follows that any quaternion can be resolved into the parts q = w -\- xi -\- yj -\- zk. These hypernumbers can easily be made the base of the whole system of quaternions, and it is one of the many methods of deriving them. Hamilton started from these. The account of his invention is contained in a letter to a friend, which should be consulted. (Philosophical Maga- zine, 1844, vol. 104, ser. 3, vol. 25, p. 489.) 6. Product of Vectors. It becomes evident at once if we consider the product of two vector parts of quaternions, or two quaternions whose scalar parts are zero, that we may consider this product, a quaternion, as the product of the vector lines which represent the vector parts of the quaternion factors. From this point of view we ignore the biradials completely, and look upon every geometric vector as the representative of the vector part of a set of quaternions with different scalars, among which one has zero scalar. From the biradial definition we have VqVr= S-VqVr+ V-VqVr equal to the quaternion whose biradial consists of two vectors in the same plane as the vector normals of the Fig. 16. 102 VECTOR CALCULUS biradials of Vq, Vr and perpendicular to them respectively. In the figure the biradial of Vr is OAB, and of Vq is OBC, and of VqVr is OAC. If then we represent the vectors by Greek letters whether meant to be considered as lines or as vector quaternions, a = Vq, /3 = Vr, then the quaternion which is the product of a(3 has for its angle the angle be- tween /3 and a + 180°, and for its normal the direction OB. If we take UVa(3 in the opposite direction to OB, and of unit length, so as to be a positive normal for the biradial a /3 in that order, then we shall have, letting 6 be the angle from a to /3, a(3 = TaTj3(- cos + UVafi sin 0). We can write at once then the fundamental formulae S-a& = - TaTfi cos 6, V-a$ = TaTp-sm 6- UVaP. From this form it is clear also that any quaternion can be expressed as the product of two vectors, the angle of the two being the supplement of that of the quaternion, the product of their lengths being the tensor of the quater- nion, and their plane having the unit vector of the quater- nion as positive normal. If now we consider the two vectors a and to be resolved in the forms a = ai-\- bj + ck, (3 — li + mj + nk, where i, j, k have the significance of three mutually tri- rectangular unit vectors, as above, then since Ta Tfi cos 6 = al-\- bm-\- en, and since the vector Ta T(3 sin 6 • UVa(3 is (bn — cm)i + (cl — an)j + (am — bl)k, we have a/3 = — (al + bm + en) + (bn — cm)i + (cl — ari)j -\- (am — bl)k. VECTORS IN SPACE 103 But if we multiply out the two expressions for a and distributively, the nine terms reduce to precisely these. Hence we have shown that the multiplication of vectors, and therefore of quaternions in general, is distributive when they are expressed in terms of these trirectangular systems. It is easy to see however that this leads at once to the general distributivity of all multiplications of sums. 7. Laws of Quaternions. We see then that the addition and multiplication of quaternions is associative, that addition is commutative, and that multiplication is dis- tributive over addition. Multiplication is usually not commutative. We have yet to define division, but if we now consider a biradial as not being geometric but as being a quaternion quotient of two vectors, we find that P/a differs from a(3 only in having its scalar of opposite sign, and its tensor is T(3/Ta instead of TaTfi. It is to be noticed that while we arrived at the hyper- numbers called quaternions by the use of biradials, they could have been found some other way, and in fact were so first found by Hamilton, whose original papers should be consulted. Further the use of vectors as certain kinds of quaternions is exactly analogous, or may be considered to be an extension of, the method of using complex numbers instead of vectors in a plane. In the plane the vectors are the product of some unit vector chosen for all the plane, by the complex number. In space a vector is the product of a unit vector (which would have to be drawn in the fourth dimension to be a complete extension of the plane) by the hypernumber we call a vector. However, the use of the unit in the plane was seldom required, and likewise in space we need never refer to the unit 1, from which t^e vectors of space are derived. On the other hand, just as in the plane all complex numbers can be found as the ratios 104 VECTOR CALCULUS of vectors in the plane in an infinity of ways, so all quater- nions can be found as the ratios of vectors in space. All vectors are thus as quaternions the ratios of perpendicular vectors in space. And multiplication is always of vectors as quaternions and not as geometric entities. In the common vector systems other than Quaternions, the scalar part of the quaternion product, usually with the opposite sign, and the vector part of the quaternion product, are looked upon as products formed directly from geometric con- siderations. In such case the vector product is usually defined to be a vector in the geometric sense, perpendicular to the two given vectors. Therefore it is a function of the two vectors and is not a number or hypernumber at all. In these systems, the scalar is a common number, and of course the sum of a number and a geometric vector is an impossibility. It seems clear that the only defensible logical ground for these different investigations is that of the hypernumber. It is to be noticed too that Quaternions is peculiarly applicable to space of three dimensions, because of the duality existing between planes and their normals. In a space of four dimensions, for instance, a plane, that is a linear extension dependent upon two parameters, has a similar figure of two dimensions as normal. Hence, corre- sponding to a biradial we should not have a vector. To reach the extension of quaternions it would be necessary to define triradials, and the hypernumbers corresponding to them. Quaternions however can be applied to four dimensional space in a different manner, and leads to a very simple geometric algebra for four-dimensional space. The products of quaternions however are in that case not sufficient to express all the necessary geometrical entities, and recourse must be had to other functions of quaternions. VECTORS IN SPACE 105 In three-dimensional space, however, all the necessary ex- pressions that arise in geometry or physics are easily found. And quaternions has the great advantage over other systems that it is associative, and that division is one of its processes. In fact it is the most complex system of numbers in which we always have from PQ = the conclusion P = 0, or Q = 0.* 8. Formulae. It is clear that if we reverse the order of the product ce/3 we have 0a = Soft - Vafi. This is called the conjugate of the quaternion a(3, and written K-a(3. We see that SKq = Sq= KSq, VKq = - Vq = KVq. Further, since qr = SqSr + SqVr + SrVq + VqVr, we have K-qr= SqSr - SqVr - SrVq + VrVq = KrKq. From this important formula many others flow. We have at once K-qi- • -q n = Kq n > • >Kqi. And for vectors Koli- • -0L n = {—) n a n - • •«!. Since Sq = i(q+Kq), Vq=\{q-Kq), we have therefore S-OLl" 'Qt2n = i(<*l" * -«2n + «2n ' * 'Oil), S-ai- • -C^n-l = i(tti* • 'tt2n~l — «2n-l' * 'Oil), V'CXi- ' 'OL 2n = !(«!« * ,Q; 2n ~ « 2n ' " '«t), F'Qfi- • ■a 2 n-l = %(<Xi' ' -OL2n-\ + «2n-l ' * -«l). * Consult Dickson: Linear Algebras, p. 11. 106 VECTOR CALCULUS In particular 2Sa$ = aft + Pa, 2SaPy = afiy - y(3a, 2Vap = a/3 - fa 2Va(3y = a(3y + y(3a. It should be noted that these formulae show us that both the scalar and the vector parts of the product can them- selves always be reduced to combinations of products. This is simply a statement again of the fact that in quaternions we have'only'one kind of multiplication, which is distributive and associative. We see from the expanded form above for S • qr that S-qr = S-rq. Hence, in any scalar part of a product, the factors may be permuted cyclically. For instance, S-afi = S-(3a, S-a(3y = S-Pya = S-yaQ, S-a(3y5 = SPyfa From the form of Sq=Uq+Kq), Sq = SKq; hence we have Sa(3 = S@a, Safiy = - Syfa Sa(3y8 = S8y(3a, etc. From the form of VKq = — Vq we see that Vafi = - V@a, Vafiy = VyPa, Vapyh m - Vdypa, Vapyhe = VebyPa- We do not have a simple relation between V-qr and V-rq, but we have the fact that they are respectively the sum and the difference of two vectors, namely, If a — SqVr -+- SrVq, P = VVqVr, then ft is perpendicular to a, and Vqr = a + P, Vrq = a — (3. q = w + w? + , ;V + ^ (Tq) 2 = w 2 + x 2 + 2/ 2 + * 2 , £g = », (TTg) 2 - z 2 + f+z\ VECTORS IN SPACE 107 It is obvious that TVqr = TVrq and that /.qr = /rq - tan -1 TVqr/Sqr. The planes differ. The product of g and i£g is the square of the tensor of q. We indicate the unitary part of q, called the versor of q, by Uq. We have then the formulae Kq = w — ix — jy — kz, j j = w + ix + jy + kz q Tq Vq = ix + jy + kz, TTVn _ix + jy + fa ^ Kg rrg ' (TVUq) 2 m (X* + f + * 2 )/(w 2 + a? + 2/ 2 + z 2 ), cos- Z g = w/Tg = #• £7g, sin-Z g = TVq/Tq= TVUq, Z-q= tan" 1 rFg/.Sg. The product of two quaternions is qr = ww* — xx' — yy' — zz' -f i(wx' -\- w'x + yz f — y'z) + j(wy' + w'y + zx' — z'x) + k(wz' + w'a + xy f — x'y). From the formula Tqr = TgTY we have a noted identity (ief +a*+ y 2 + z 2 ) <>' 2 + x' 2 +y' 2 + s' 2 ) = (ww f — ao' — 2/2/' — zz') 2 + (wa;' + w'x + 2/2' — S/'s) 2 + (wy' + to'y + zx' — z'x) 2 + (W + w'z + #2/' — ^'2/) 2 - This formula expresses the sum of four squares as the product of the sums of four squares. It was first given by Euler. The problem of expressing the sum of three squares as the product of sums of three or four squares and the sum of eight squares as the product of sums of eight squares has also been considered. 108 VECTOR CALCULUS 9. Rotations. We see from the adjacent figure that we have for the product qrq- 1 a quaternion of tensor and angle the same as that of r. But the plane of the product is produced by rotating the plane of r about the axis of q through an angle double the angle of q. In case r is a vector /3 we have as the product a vector fi f which is to be found by rotating conically the vector (3 about the axis of q through double the angle of q. It is obvious that operators* of the type qQq~ l , r()r -1 , which are called rotators, follow the same laws of multiplica- tion as quaternions, since g(r()r _1 )<7 -1 = qrQ[qr]~ l . A gaussian operator is a rotator multiplied by a numerical multiplier, and is called a mutation. The sum of two mutations is not a mutation. As a simple case of rotator we see that if q reduces to a vector a we have as the result of after 1 = /3' the vector which is the reflection of /3 in a. The reflection of /3 in the plane normal to a is evidently — a$or l . EXAMPLES (1) Successive reflection in two plane mirrors is equivalent * QOq' 1 represents a positive orthogonal substitution. VECTORS IN SPACE 109 to a rotation about their line of intersection of double their angle. (2) Successive reflection in a series of mirrors all per- pendicular to a common plane, 2h in number, making angles in succession (exterior) of <pu, (P23, <&*••• is equivalent to a rotation about the normal to the given plane to which all are orthogonal, through an angle 6 = 2h — ir — 2(<p 12 + (pu + ••• + <P2h-i,2h) which is independent of the alternate angles. (3) Study the case of successive reflections in mirrors in space at any angles. (4) The types of crystals found in nature and possible under the laws that are found to be true of crystals, are solids such that every face may be produced from a single given face, so far as the angles are concerned, by the following op9rations : I, the reversal of a vector, in quaternion form — 1 . A, rotation about an axis a a n ()oT n . L4, rotatory inversion about a — a n ()a~ u . S, reflection in a plane normal to /5 — jSO/S -1 = /?()/?. The 32 types of crystals are then generated by the succes- sive combinations of these operations as follows: Triclinic Ci Asymmetric 1. d Centre-symmetric 1,-1. Monoclinic C s Equatorial 1, 0Q0. d Digonal polar 1, a()a -1 . C 2 h Digonal equatorial 1, a()a; -1 , a()a. Orthorhombic C 2v Didigonal polar 1, a()a~ l , 0Q0, Sap = 0. D 2 Digonal holoaxial 1, a()a-\ fiQfi' 1 , Sap = 0. Du Didigonal equatorial .... 1, a()a -1 , POP' 1 , «()«, SaP = 0, A = a l ' 2 0a- 1 ' 2 . Tetragonal d Tetragonal alternating . .1, — A. Du Ditetragonal alternating. 1, — A, P{)P~ X . d Tetragonal polar 1, A. 110 VECTOR CALCULUS Ctk Tetragonal equatorial. . .1, A, aQa. C4* Ditetragonal polar 1, A, /3()/3. D4 Tetragonal holoaxial .... 1, A, 0Q0~K Dak Dietragonal equatorial . . 1, A, aQa, /3()/3 _1 . Rhombohedral C 8 Trigonal polar l,B, where B is a 2l3 0<*~ il3 ' Czi Hexagonal alternating . .1, B, — B. Ctv Ditrigonal polar 1, B, pQ0. • D, Trigonal holoaxial 1, B, 0Q0T+. Did Dihexagonal alternating . 1, B, j8()/8~ l , 7O7, 7 bisects Z/3, B0. Hexagonal Czh Trigonal equatorial 1,5, aQa. Dzh Ditrigonal equatorial . . .1, B, aQa, jS()/3 _1 . d Hexagonal polar 1, C, where C = a 1/3 ()«~ 1/3 . dh Hexagonal equatorial . . . 1, C, aQa. Civ Dihexagonal polar 1, C, /3()yS, where Sap = 0, bisects angle of 7 and Cy, Say = 0. Di Hexagonal holoaxial .... 1, C, /3()/S _1 . Dan Dihexagonal equatorial. .1, C, a()a, pQ(3~ l . Regular T Tesseral polar ..1, aQa' 1 , PQP~ X , Safi = Spy = Sya = 0, L where L = (a + fj + 7)0(«+/3 + 7 )- 1 . T h Tesseral central 1, aQa~\ 0Q/T 1 , 7O7" 1 , L, aQa. T d Ditesseral polar 1, aQa' 1 , 0Q0-\ 7O7" 1 , L, (a + fi)Q(a + /3). Tesseral holoaxial 1, aQa-\ 0Q0~ l t yQy~ l , L, (a + p)Q(a + P)~K Oh Ditesseral central 1, aQa~\ $00-*, yQy' 1 , t, t {« + 0)Q(a+0? t aQa. The student should work out in each case the fuJl set of operators and locate vectors to equivalent points in the various faces. Ref. — Hilton, Mathematical Crystallography, Chap. IV- VIII. (5) Spherical Astronomy. We have the following nota- tion: X is a unit vector along the polar axis of the earth, h is the hour-angle of the meridian, VECTORS IN SPACE 111 L = cos h/2 + X sin h/2, i = unit vector to zenith, j = unit vector to south, k = unit vector to east, X = i sin I — j cos /, where I is latitude, li = unit vector to intersection of equator and meridian, \x — i cos I -\- j sin I, aSX/x = SkX = Sk/j, = 0, d = declination of star, 5 = unit vector to star on the meridian = X sin d + jjl cos d, z = azimuth, A = altitude. At the hour-angle h, 8 becomes 8' = L~ l 8L. The vertical plane through 8 f cuts the horizon in iVi8' = JSJ8' + kSk8', tan z = Sk8'/Sj8'. At rising or setting z is found from the condition Sid' = 0. The prime vertical circle is through i and k. The 6-hour circle is through X and V\ji. a — right ascension angle, t = sidereal time in degrees, h = t + a, L t = cos t/2 + X sin t/2, L a = cos a/2 + X sin a/2, e = pole of ecliptic, X = first point of aries = vernal equinox = Lr l ^L t} s = longitude, b = latitude, M = cos s/2 + e sin s/2. Problems. Given /, d, find A and z on 6-hour circle. Sfx8' = 0. /, d, find h and z on horizon. /, d, find A. I, d, A, find h and z, 8' = L~ l 8L = i cos A + ? cos s + k sin 2. 112 VECTOR CALCULUS /, d, h, find A and z. a and d, find s and b. (G) The laws of refraction of light from a medium of index n into a medium of index n' are given by the equation nVvct — n'Vva! where v, a, a' are unit vectors along the normal, the incident, and the refracted ray. The student should show that Investigate two successive refractions, particularly back into the first medium. (7) It is easy to show that if q and r are any two quater- nions, and /3 = V • VqVr, we may write (8) For any two quaternions qiq' 1 ± r _1 ) = (r =b q)f\ and = r(r ± q)~ l q. -± - 9 r (9) If a, b, c are given quaternions we can find a quater- nion q that will give three vectors when multiplied by a, b, c resp. That is, we can find q, a, ft y such that aq = a, bq = ft eg = 7. (R. Russell.) We have a — — V • Vc/aVa/b, etc., or multiples of these. (10) In a letter of Tait to Cayley, he gives the following: (q+ r)()(g+ r)" 1 = (qlr) x rQf-i( q /r)-* = qiq-iryQiq-^-vq- 1 = qh^Qq-^q- 1 , (Vq+ Vr)()(Vq+ Fr)" 1 = fa/rWJf^fo/r)- 1 /*, VECTORS IN SPACE 113 where tan xA = a sin A/ (a cos A + 1), c sin 2la sin ra/3 + cos 2la cos ra/3 = 2 (a cos o + & cos /S) V (6 sin |8), 2c + sin 2/a cos ra/3 = 2a sin a/ (6 sin /3). Interpret these formulae. 10. Products of Several Quaternions. We will develop some useful formulae from the preceding. If we multiply a(3-(3a we have a 2 (3 2 - S 2 a(3 - V 2 a(3. Since Sax = 0, if x is a scalar, &*/3t = SaVfry, Sa(3y8 = SaVfiyb, etc. Since 2Va(3 = a(3 - (3a, 2Sa(3 = a(3 + 0ce, ffiaV(3y = af3y — ay (3 — (3ya + 7/fa = 2(7/3o — 07/?) = 2(y(3a + 7«/3 — ay {3 — yap). For 2<S/?7 • a = /57a: + 7/fo = 2aSj3y = 0:187 + 0:7/?, whence 0:187 — $70 = Yj8o — 07/?. Therefore VaV(3y = ySa(3 - (3Say. Adding to each side ccSfiy, we have Va(3y = aS(3y - (3Sya + ySa(3. Since ]S = crtaft = a^SaP + a~Wa$, which resolves (3 along and perpendicular to a, Sqrq -1 = Sr = qSrq -1 , Vqrq- 1 = h^q~ l - Kq~ l KrKq) = iC^a -1 — qKrq~ l ) = qVr-q~ l . That is, if we rotate the field, Sr and TTr are invariant. 114 VECTOR CALCULUS Hence Vapy = VafiyaoT 1 = aV(3ya-oT l and Vafty, Vfiya. are in a plane with a and make equal angles with a. For instance if a, /?, y, Vafly, Vfiya, Vyafi intersect a sphere, then a, /?, y bisect the sides of the triangle Vafiy, Vpya, Vya(3, a being opposite to Vya(3, etc. Evidently if «i, (X2- • -a n are n radii of a sphere forming a polygon, then they bisect the sides of the polygon, given by Vaia 2 - • -a n , F«2«3- • '<x n , Vets- - -a n aia2, • • -Va n (xi- - -a n -i. This ex- plains the geometrical significance of these vectors. In fact for any vector a and quaternion q, the vector a bisects the angle between Vqa and Vaq, that is to say we construct Vqa from the vector Vaq by reflecting it in a. The same is true for any product, thus (3yde • • • vol is different from a(3y8e • • • v only in the fact that its axis is the reflection in a of the axis of the latter. <M3 ' ' ' Qnqi differs from qiq 2 • • • q n only in the fact that its axis has been rotated negatively about the axis of q\ through double the angle of qi. Indeed ?2?3- • -q n qi = q~Kqiq2- ■ -qn)q\. If we apply the formula for expanding VaVfiy to V(Vafi)Vy8 = — V(Vy8)Va(3 we arrive at a most im- portant identity: V-VapVy8 = 8Sa$y - ySa$8 = - V-VydVafi = aS(3y8 - /3Say8. From this equality we see that for any four vectors 8Sapy = aSfiyd + @Sya8 + ySaj38. This formula enables us to expand any vector in terms of any three non-coplanar vectors. Again 5Sapy - VpySad = V-aV(V(3y)8 = - V-aV8V$y = Fa(3Sy8 - VayS(38. VECTORS IN SPACE 115 We have thus another important formula SSofiy = Va(3Sy5 + VfiySaB + VyaS08, enabling us to expand any vector in terms of the three normals to the three planes determined by a set of three vectors, that is, in terms of its normal projections. Since aSPyS = VpySad + VytSefi + VbfiSay and (3Syda = Vay S{38 + VySSofi + VdaSPy, we have VVapVyd = VabSPy + VPySad - VayS(38 - VpbSay. From this we have at once an expansion for Vafiyh, namely Vctfyd = Va(3Sy8 - VaySpb + VabSPy + SapVyb - SayVpb + SabVpy. Also easily Sapyd = SaPSyd - SaySpd + SabSpy. SVapVyb = SadSPy - SaySpb. V-ap-Sybe = yS-VapVbe - bS-VapVye + eS- VapVyb y b e Say Sab Sae SPy SPB Spe In the figure the various points lie on a sphere of radius I. The vectors from the center will be designated by the corresponding Greek letters. The points X, Y, Z are the midpoints of the sides of the A ABC. From the figure it is evident that H» = yli = (7/« 1/2 , v/y - «h m (a/7) 1 *, Whence 7 = sar 1 , « - nrr\ p = ^r 1 , 116 VECTOR CALCULUS P Fig. 18. where v = it 1 !, and the axis of p is ± a. Also p%p~ l = ^iT 1 ^ 7 ? -1 * so that if P is the pole of the great circle through XY then the rotation pQp~ l brings £ to the same position as the rotation around OP through twice the angle of tjJ -1 . Since £ goes into {' by a rotation about OA as well as one about 0P f this means that the new position 0Z r is the reflection of OZ in the plane of OP A. The angle of p is then ZAL or ZAP according as the axis is -\- a or — a. The angles of L and M are right angles, and if we draw CN perpendicular to XY then ANCY = ALAY, ANCX = AMBX, and AL = BM = CN and APB is isosceles. Hence the equal exterior angles at A and B are ZAL = ZBM = \{A + 5 + Q. Draw PZ, then /ZiM = Zv^ 1 for it =JzJWM = \ML = ZF since ilfZ - XN and iVF = YL. The angle between the planes LAP and ZOP is thus the biradial 7)%~ l and also £" is the biradial whose angle is that of the VECTORS IN SPACE 117 planes OAZ, ZOP, so that ZOA and AOL make an angle equal to z p, hence ZV = h(A + B+C). Further pa' 1 - nlyyfc'tla = («/t) 1/2 (t/« 1/2 (/3/«) 1/2 - p'. The angle of p' is thus %(A + 5 + C - tt) = 2/2 where S is the spherical excess of AABC. Consider the quaternion p = r)^ 1 ^ = — 77^". The con- jugate of p is Kp = ££77, whose axis is also a and angle - \{A + B + 0). Thus the quaternion ffij = - sin 2/2 - a: cos 2/2. Shifting the notation to a more symmetric form we have for any three vectors aia 2 as = — sin 2/2 — TJVai(x 2 a.z • cos 2/2 = cos \<j — k sin Jo - , where 2 is the spherical excess of the triangle the midpoints of whose sides are A\, A 2 , A% and a is the sum of the angles of the triangle. Hence Saia 2 a 3 = cos Jo", Va ia 2 a 3 = ~* UV<x\ol 2 ccz sin \a. It is to be noted that the order as written here is for a positive or left-handed cycle from A\ to A 2 and A$. Since 2 is the solid angle of the triangle, — S-a\a 2 as is the sine of half the solid angle and — TVa\a 2 az is the cosine of half the solid angle, made by oi, a 2 , a 3 . If now we have several points as the middle points of the sides of a spherical polygon, say aia 2 - • -a n and the vertex between a\ and a n is taken as an origin for spherical arcs drawn as diagonals to the vertices of the polygon, then for the various successive triangles if we call the midpoints of the successive diagonals J*lj $2, ' ' "fn-3 118 VECTOR CALCULUS we have, taking the axis to the origin which we will call k, and which is the common axis of all the quaternions made up by the products of three vectors The sum of the angles of the polygon is the sum of the angles of all the triangles into which it is divided, so that if this sum is a we have for any spherical polygon «i«2- • *«n = (— ) n_3 [cos cr/2 — k sin a/2]. We are able to say then that if the midpoints of the sides of a spherical polygon are ai, a 2 , • • -a nt then SoCi(X2' ' '0i n = db COS ff/2, where a is the sum of the angles ; the vertices of the polygon are given by Wolioli- • -a n , TJVcioOLz - • • a n ai, ••'•, UVa n - • -ttn-l, each being the vertex whose sides contain the first and last vectors in the product; and the tensors of these vectors are each equal to sin <r/2. The expression — Sa(3y is called the first staudtian of afiy, the second staudtian is - SVapVPyVya/TVapTVPyTVya = S 2 aj3y/TVaPTV(3yTVya, which is evidently the staudtian of the polar triangle. S-ai--a n ,i r i i mrz — — — • = tan f solid angle. 1 V •«!• • -a n We will summarize here the significance of the expressions worked out thus far, and in particular the meaning of their vanishing. VECTORS IN SPACE 119 Sa(3 is the product of TaTp by the cosine of the angle between a and — 0. It vanishes only if they are per- pendicular. Vafi is the vector at right angles to both a (3 whose length is TaTfi multiplied by the sine of their angle. It vanishes only if they are parallel. Safiy is the volume of the parallelepiped of a fi y, taken negatively. It vanishes only if they are all parallel to one plane. Vafiy, Vafiyd, • •'• these vectors are the edges of the poly- hedral giving the circumscribed polygon, and if the ex- pression vanishes, we have by separating the quaternion, Va0y8- • • = aS(3y8- • • + VaVPyS-'- = 0. Hence a is the axis of (3yd- • • and Sfiyd- • • equals zero. By changing the vectors cyclically we have n vectors all of which have a zero tensor, so that each edge is the axis of the quaternion of the other n — 1 taken cyclically. This quaternion in each case has a vanishing scalar. n = 3, a j8 y are a trirectangular system. n = 4, a (3 y 8 are coplanar, shown by the four vanish- ing scalars. The angle a(3 = angle 7#. n = 5, the edge Va(3y is parallel to V8e and cyclically similar parallelisms hold. We have in all these cases the sum of the angles of the circumscribing polygon a multiple of 2w and it satisfies the inequality S(n — 2)tt is greater than a which is greater than {n — 2)x. It is evident that if the polygon circumscribed has 540° the vectors lie in one plane. ■ Safiyb = 0. If e = Va(3y8, then VaQySe = 0, and the preceding case is at hand for the five vectors. S-aia 2 - • -oL n = 0, the sum of the angles of the polygon is an odd multiple of x. 120 VECTOR CALCULUS EXERCISES 1. S-VaPVpyVya = - (Sapy)* V-VapVpyVya = VaP(y*SaP - SPySya) + ..... 2. S(a + P)iP + 7)(7 + «) m 2Sa0y. 3. 5-F(a + /3)(0 + 7)708 + 7)(7 + a)V(y + «)(a + 0) 4. 5.F(Fa/3F/37)(F/37^7«)7(F7aFa/3) = - (S-afiy)*. 5. S-5ef - - 16(5 -a^) 4 , where 5 = F(F[« + 0[\fi + 7]F[^ + 7 ][7 + «]), < = y(7D9 + 7][7 + a]V[y + a][a + fl), f = V(V[y + «][a + /S]7[a + 0]\fi + 7]). 6. S(xa + yP + 27 )(x'a + y'0 + *'7)(x"a + y"0 + *"7) 4(5.afl 7 ) 1 . 7. x \ X' X" S-aiPiyi = - Saai Sftai Syai Sa0i S00i Syfii Say 1 Sfiyi Syyi Saai Sa&i S0 ai sm Syai Syffi S8ai S8P1 s s s ,8 ayi Sadi Pyi SP81 771 Sy8i 571 S881 ■■ S • a/37. for any eight vectors. If the element Saai is changed to Szai the value is - S-0y8'S'Piyi8i-S-ai(e — a). 9. S-Va0yV0yaVyaP = ISaPSPySyaSaPy. 10. From S 2 P /a - V 2 P fp = 1 we find T(Sp/a + Vp/P) where 1 = T{\cl+ p + \pa~i - irv + yr 1 ) = T(a' P + p/80 a' = §(«T* - r>), pV = i(a"» + p*-«). 11. If T P = Ta = Tp = 1 and S-afip = 0, S-U(p-a)U(p -P) = ±iV[2(l -Sap)]. 12. If a, P, 7 and a h Pi, 71 are two sets of trirectangular unit vectors such that if a = Py, a x = Piy, then we may find angles called Eulerian angles such that a 2 = a COS yp + P sin \J/, P 2 = — a sin i£ + P COS ^, 73 = 7 cos 6 + <*2 sin 0, a 3 = — 7 sin -f « 2 cos 0, 71 = 73, «i = «3 cos ^ + /?2 sin <p, Pi = — a 3 sin v> + /3 2 cos <p. VECTORS IN SPACE 121 13. If q = ai«2 • • • ot n then if we reflect an arbitrary vector in succession in a„, a n -i, • • • 0:20:1 when Sq = the final position will be a simple reflection of p in a fixed vector, and if Vq = the final position will be on the line of p itself. Similar statements hold if the reflections are in planes that are normal respectively to a n , • • • «i. 11. Functions. We notice some expressions now of the nature of functions of a quaternion. We have the follow- ing identity which is useful : (a/3) n + {$a) n = (ol$ + $a)l(<xP) n ~ l ] ~ a^a[(a^ n ~ 2 = 2SaP[(a(3) n ~ 1 + (/to) 71-1 ] - a 2 ^[(a^ n ~ 2 + 08*)*-*]. Whence 2 n S n a(3 = (a/3 + M n = [(«/3) n + (fax)"] + lt/ nl ni K«/5)"- 2 + (/3a)"" 2 ]a 2 /3 2 \\{n — 1)1 + 2l(n w l 2)1 [(«» n_4 + w-v/3 4 + • • • \\{n — 1)1 This implies the familiar formula for the expansion of cos n in terms of cos nd, cos (n — 2)0, and we can write as the reverse formula S(a(3) n - (-) w / 2 [a n /3 n - n 2 S 2 a(3-a n - 2 l3 n - 2 l2\ + n 2 (n 2 - 2 2 )SV-a n " 4 /3"- 4 /4! - • • •] n even (- ) (n ~ l) ' 2 [nSa(3 • a n ~ l er- l ll ! - n(n 2 - l 2 )5 3 a/5-o: n - 3 /3 n - 3 /3! + • • •] n odd. Likewise TV 2n a$= (-l) n /2 2n - 1 [S(al3 2n (2n)! l!(2n- 1) S(aP 2n ~ 2 a 2 p 2 + ■••] 122 VECTOR CALCULUS 7»p»-i a/3== (_l)«/2 2 «- 2 [7T(a/3) 2n - 1 - ( 2n - ^ l TV(aB) 2n ~ 3 + • . .1 l!(2n-2)1 1VKfxp) x J TV(ap) n /TVap = (-) n/2 [n5a i S-Q: n - 2 /? n - 2 /l! - n(n 2 - 2 2 )iS 3 a/3« n-4 /S n ~ 4 /3! + • • -J n even (_1)<*-*^1 - (n 2 - l^SPap-cT+p^fil + • • •] n odd. Since jS/a is a quaternion whose powers have the same axis we have (1 — 0/a) -1 = 1 + fi/a + 03/a:) 2 + • • • when Tfi < Ta, and taking the scalar gives the well-known formula Likewise S-^~= 1 + S/5/a + S(/3/a) 2 + a — p TV-^—= TVp/a + TV(p/a) 2 + a — p If we define the logarithm as in theory of functions of a complex variable we have log (1 - fi/a) = log 7(1 - fi/a) + log 17(1 - fi/a) = - fa - Itf/a)* - HP/a)* . Therefore log f(l - fi/a) - - Sfi/a - §S(/?/c*) 2 Z °LZ_1 = TV log (1 - fi/a) = TVp/a + ^TV(p/a) 2 - a Again T{a - p)~ l = Ta' 1 - f(l - P/a)- 1 - fo^l + Pi(- SUp/a) TP/a + P 2 (- SUp/a) T 2 P/a + .••], where Pi P2 are the Legendrian polynomials. Evidently for coaxial quaternions we have the whole theory of functions of a complex variable applicable. VECTORS IN SPACE 123 12. Solution of Some Simple Equations. (1). If ap = a then p = oT l a. (2) . If Sap = a then we set Vap = f where £* is any vector perpendicular to a, and adding, p = aa _1 + a~ l $. (3). If Fap = jS then *Sap = a: where # is any scalar, and adding we have p = a~ l (3 + aaaf" 1 . (4). If Vapfi = y then SaVapQ = &x 2 p/3 = <* 2 £p/3 = Say and SpVap(3 = /3 2 £ap = S/fy. Now Fap/5 = aS/3p - pSafi + (3Sap and substituting we have p = [o;- 1 ^7 + /T 1 ^ - y]/8afi. The solution fails if Sa(3 = 0. In this case the solution is p = _ a-'S^y - p^Sa-iy + xVofi, x any scalar. (5). If Yapp = 7 then Sa(3pSafi = &*07 and Soft) = Sa(3y/Sa(3. Adding to Va(3p, we have afip = 7 + Sa(3y/SaP and p = 0^or*7 + '(hcT*8cfiyl8c&. (6). If &xp = a, £/3p = b, then a^p = zFa/3 + V(al3 - ba)Va(3. (7). If Sap = a, S(3p = b, Syp = c, then pSafiy = aV(3y + bVya + cFa/5. (8). If gag -1 = |3 then g = (x/3 + y)/(a + /3) where x and ?/ are any scalars. Or we may write q = u + 0(a + |8) + wFa?/3 where u = — w#a(a: + /3). (9). If gag" 1 = y, q^q' 1 = 8, then V(y - a)(8 - ft! .. 1 + S(T + «)(«- ft 124 VECTOR CALCULUS (10). If qaq- 1 = f, qpq~ l = *, qyq~ l = f, then S-flft - «) - 0, S-q( V - ft - 0, flf.gtf - 7) = 0, hence Fg is coplanar with the parentheses, and we have x(i - a) + 2/(77 - ft + H(f - 7) = where »:*:*->- 2S 7 (r? - ft : 2Sy(i - a) : S(£ + a)(i, - ft. The six vectors are not independent. Vq is easily found and thence Sq from qa = £q. (11). If (p - a)" 1 + (p - ft" 1 - (P ~ 7)" 1 ~ (P ~ 5)- 1 = 0, then if we let ifi' ~ aT 1 = 1 * (TO - 5)" 1 - 5] - [(a - 6)" 1 - 5]) = (p — 8)(p — a) _1 (« — 5), etc., where p', a', 0', 7' are the vectors from D, the extremity of 5, to the inverses with respect to D, of the extremities of p, a, ft 7, then (p' - a')" 1 + (p' - ft)" 1 - (p' - 7T 1 = 0. Prove that 1 - ft _ y - ft _ P ' - y _ r y - /n i/2 p whence p' and p. (R. Russell.) (12). If (q - a)" 1 + (q - 6)" 1 - (q - c)" 1 - (q - d)~' = 0, we set (q - d)(q' - d)= (a- d){a' - d) = (b - d)(b' - d) = (c - d){c' - d) - 1, VECTORS IN SPACE 125 thence (q - d)-> -(q- a)' 1 = (4 - d)-\a - d)(q - a)' 1 (q - d)~i [(a~d)/(q-a)+(b- d)l(q- b)- (c - d)/(q- e)] - (?' - a')' 1 + (?' - &T 1 ~ W ~ cT 1 and we have q' from (V - cW - C) = (g' - 6')/(g' - «0 = (q' - c')l(a' - c') - [(V - c')Ka' - c')]K (R. Russell.) 13. Characteristic Equation. If we write q = Sq + Vq and square both sides we have q 2 = S 2 q + (Vq) 2 + 2Sq-Vq whence g 2 - 2qSq + S 2 q - V 2 q = 0. This equation is called the characteristic equation of q. The coefficients 2Sq and S 2 q - V 2 q = T 2 q are the invariants of q; they are the same, that is to say, if q is subjected to the rotation r()r -1 . They are also the same if Kq is substituted for q. Hence they will not define q but only any one of a class of quaternions which may be derived from each other by the group of all rotations of the form rQr~ l or by taking the conjugate. The equation has two roots in general, Sq + Tqyl - 1 and Sq - Tq^ - 1. Since these involve the V — 1 it leads us to the algebra of biquaternions which we do not enter here, but a few re- marks will be necessary to place the subject properly. Since the invariants do not determine q we observe that we must also have UVq in order to have the other two parameters involved. 126 VECTOR CALCULUS If we look upon UVq as known then we may write the roots of the characteristic equation in the number field of quaternions as Sq + TVqUVq and Sq — TVqUVq or q and Kq. If we set q -f- r for q and expand, afterwards drop all the terms that arise from the identical equations of q and r separately, we have left the characteristic equation of two quaternions, which will reduce to the first form when they are made to be equal. This equation is qr+rq-2Sq-r- 2Sr-Vq + 2SqSr - 2SVqVr = 0. We might indeed start with this equation and develop the whole algebra from it. We may write it qr-\- rq- 2qSr - 2rSq + 4Sq-Sr + S-qr + S-rq = which involves only the scalars of q, r, qr, and rq. 14. Biquaternions. We should notice that if the param- eters involved in q can be imaginary or complex then division is no longer unique in certain cases. Thus if Q 2 =q 2 we have as possible solutions Q = ± q and also Q = ± V (- l)UVq-q. If q 2 = and Vq = then TVq = and we have Vq = x(i + j V — 1) where X is any scalar and i, j are any two perpendicular unit vectors. CHAPTER VII APPLICATIONS 1. The Scalar of Two Vectors 1. Notations. The scalar of the product of two vectors is defined independently by writers on vector algebra, as a product. In such cases the definition is usually given for the negative of the scalar since this is generally essentially positive. A table of current notations is given. If a and (3 define two fields, we shall call S*cfi the virial of the two fields. S-a(3 = — a X /3 Grassman, Resal, Somoff, Peano, Bura- li-Forti, Marcolongo, Timerding. — Cfft Gibbs, Wilson, Jaumann, Jung, Fischer. — a/3 Heaviside, Silberstein, Foppl, Ferraris, Heun, Bucherer. — (aft) Bucherer, Gans, Lorentz, Abraham, Henrici. — a|/3 Grassman, Jahnke, Fehr, Hyde. Cos a/3 Macfarlane. [a/3] Caspary. For most of these authors, the scalar of two vectors, though called a product, is really a function of the two vectors which satisfies certain formal laws. While it is evident that any one may arbitrarily choose to call any function of one or more vectors their product, it does not seem desirable to do so. For Gibbs, however, the scalar is defined to be a function of the dyad of the two vectors, which dyad is a real product. The dyad or dyadic of Gibbs, as well as the vectors of most writers on vector analysis, are not considered to be numbers or hypernumbers. 127 128 VECTOR CALCULUS They are looked upon as geometric or physical entities, from which by various modes of "combination" or de- termination other geometric entities are found, called products. The essence of the Hamiltonian point of view, however, is the definition by means of geometric entities of a system of hypernumbers subject to one mode of multiplica- tion, which gives hypernumbers as products. Functions of these products are considered when useful, but are called functions. 2. Planes and Spheres. It is evident that the condition for orthogonality will yield several useful equations, and of these we will consider a few. The plane through a point A, whose vector is a, per- pendicular to a line whose direction is 8 has for its equation, since p — a is any vector in the plane, S-d(p-a) = 0. If we set p = 8Sa/d we have the equation satisfied and as this vector is parallel to 5 it is the perpendicular from the origin to the plane. The perpendicular from a point B is b~ l S{a - 0)5. If a sphere has center D and radius T(3 where /? and — (3 are the vectors from the center to the extremities of a diameter, then the equation of the sphere is given by the equation S(p - 3 + fi)(p - d - P) = 0, orp 2 - 2S8 P + 5 2 - /3 2 = 0. The plane through the intersection of the two spheres p 2 - 2£5ip + ci = = p 2 - 2S8 2P + c 2 is 2S(5i — 5 2 )p = ci — c 2 . The form of this equation shows that it represents a plane APPLICATIONS 129 perpendicular to the center line of the spheres. The point where it crosses this line is X18] + x 2 8 2 P = i » Xi + x 2 whence solving, we find p = v(h + 8 2 )-\V8,8 2 + i(cj - <*)>. 3. Virial. If (3 is the representative of a force in direction and magnitude then its projection on the direction a is a~ 1 Sa^ f and perpendicular to this direction crWafi. If a is in the line of action of the force, the projection is fit If a is a direction not in the line of action then the projection gives the component of the force in the direction a. If a is the vector to the point of application of the force then Sa(3 is the virial of the force with respect to a, a term intro- duced by Clausius. It is the work that would be done by the force in moving the point of application through the vector distance a. If a fe an infinitesimal distance say, 8a, then — S8a(3 is the virtual work of a small virtual dis- placement. The total virtual work would be 8V = — 2S8a n (3 n for all the forces. 4. Circulation. In case a particle is in a vector field (of force, or velocity, or otherwise) and it is subjected to successive displacements 8p along an assigned path from A to B, we may form the negative scalar of the vector intensity of the field and the displacement. If the vector intensity varies from point to point the displacements must be infinitesimal. The sum of these products, if there is a finite number, or the definite integral which is the limit of the sum in the infinitesimal case, is of great importance. If a point is moving with a velocity a [cm./sec] in a field of force of /3 dynes, the activity of the field on the point is 130 VECTOR CALCULUS — S-(3<t [ergs/sec.]. The field may move and the point remain stationary, in which case the activity is S-(3a. The activity is also called the effect, and the power. If <r is the vector function of p which gives the field at the point P we have for the sum - 2Sa8p or - // Sa8p. This integral or sum is called the circulation of the path for the field a. 5. Volts, Gilberts. For a force field the circulation is the work done in passing from A to B. If the field is an electric field E, the circulation is the difference in voltage between A and B. If the field is a magnetic field H, then the circula- tion is the difference in gilbertage from A to B. It is measured in gilberts, the unit of magnetic field being a gilbert per centimeter. There is no name yet approved for the unit of the electrostatic field, and we must call it volt per centimeter. The unit of force is the dyne and of work the erg. 6. Gausses and Lines. In case the field is a field of flux a, and the vector TJv is the outward normal of a surface through which the flux passes, then - SaUv is the intensity of flux normal to or through the surface per square centimeter. The unit of magnetostatic flux B is called a gauss; the unit of electrostatic flux D is called a line. The total flux through a finite surface is the areal integral — fSaUvdA, written also — fSadv. The flux-integral is called the transport or the discharge. Thus if D is the electric induction or displacement, the APPLICATIONS 131 discharge through a surface A is — fSDUvdA, measured in coulombs. Similarly for the magnetic induction B, the discharge is measured in maxwells. 7. Energy-Density. Activity-Density. Among other scalar products of importance we find the following. If E and D are the electric intensity in volts/cm. and induction in lines at a point, — |$ED is the energy-density in the field at the point in joules/cc. If H and B, likewise, are the magnetic intensity in gilberts/cm., and gausses, respectively, — 2^#HB is the energy in ergs. If J is the electric cur- rent-density in amperes/cm. 2 , — S • E J is the activity in watts/cc. If G is the magnetic current-density in heavi- sides*/cm. 2 , — S ■ H G is the activity in ergs/sec. If the field varies also, the electric activity is — >S- E(J + D) and the magnetic activity — $H(G + B). EXERCISES 1. An insect has to crawl up the inside of a hemispherical bowl, the coefficient of friction being 1/3, how high can it get? 2. The force of gravity may be expressed in the form a = — mgk. Show that the circulation from A to B is the product of the weight by the vertical difference of level of A and B. 3. If the force of attraction of the earth is <r = — hUp/p 2 show that the work done in going from A to B is hiTa- 1 - T0- 1 ]. 4. The magnetic field at a distance a from the central axis of an infinite straight wire carrying a current of electricity of / amperes is H = 0.2ia -1 (— sin di + cos 6j) (i andj perpendicular to wire) and the differential tangent to a circle of radius a is ( — a sin 6 i + a cos 9j)dd. Show that the gilbertage is 0.2/ (0 2 — 0i) gilberts, which for one turn is OAirl. Prove that we get the same result for a square path. 5. The permittivity k of a specimen of petroleum is 2 [abfarad/cm.], and on a small sphere is a charge of 0.0001 coulomb. The value of the displacement D at the point p is then D = 9^2 UplTp2 [lineg] * A heaviside is a magnetic current of 1 maxwell per second. 132 VECTOR CALCULUS What is the discharge through an equilateral triangle whose corners are each 4 cm. from the origin, the plane of the triangle perpendicular to the field? 6. If magnetic inductivity p. is 1760 [henry/cm.] and a magnetic field is given by H = la [gilbert/cm.], then the magnetic induction is B = 7 -1760a [gausses]. What is the flux through a circular loop of radius a crossing the field at an angle of 30°? 7. If the velocity of a stream is given by <r = 24(cos 6 i -f sin dj), what is the discharge per second through a portion of the plane whose equation is Sip = — 12 from d = 10° to 6 = 20°? 8. The electric induction due to a charge at the origin of e coulombs is D = - eUp/T P Hir [lines]. What is the total flux of induction through a parallelepiped whose center is the origin? 9. The magnetic induction due to a magnetic point of m maxwells is B = - mUp/Tp 2 [gausses]. What is the total flux of induction through a sphere whose center is the point? 10. In problem 8, if the permittivity is 2 = k, then the electric intensity E = rH>4r. What is the amount of energy enclosed in a sphere of radius 3 cm. and center at a distance from the origin of 10 cm.? 11. In problem 9, if the inductivity is 1760 and the magnetic in- tensity is H = p~% how much energy is enclosed in a box 2 cm. each way, whose center is 10 cm. from the point and one face perpendicular to the line joining the point and the center? 12. If the current in a wire 1 mm. in diameter is 10 amperes and the drop in voltage is 0.001 per cm., what is the activity? APPLICATIONS 133 13. If there is a leakage of 10 heavisides through a magnetic area of 4 cm. 2 , and the magnetic field is 5 gilberts/cm., what is the activity? 14. Through a circular spot in the bottom of a tank which is kept level full of water there is a leakage of 100 cc. per second, the spot having an area of 20 cm. 2 . If the only force acting is gravity what is the activity? 15. If an electric wave front from the sun has in its plane surface an electric intensity of 10 volts per cm., and a magnetic intensity of 0033 gilberts per cm., and if for the free ether or for air y. = 1 and k = £-10~ 20 , what is the energy per cc. at the wave front? (The average energy is half this maximum energy and is according to Langley 4.3 -10 -5 ergs per cc. per sec.) 16. If a charge of e coulombs is at a point A and a magnetic point at B has m maxwells, what is the energy per cc. at P, any point in space, the medium being air? 8. Geometric Loci in Scalar Equations. (1). The equation of the sphere may be written in each of the forms a/p = Kp[a, S(p - a)/(p + a) = 0, S2a/(p + a) = 1, S2p/(p + <*) - 1, T(Sp/a + Vp/a) = 1, Tip - ca) m T(cp - a), S{p - a) (a - »08 - 7)(Y - B)(S - p) - 0, a 2 Sfiyp + j3 2 Syap + y 2 Sa(3p = p 2 Sa(3y (p-aO 2 (p-/3) 2 (p-7) 2 (P-5) 2 (p - a) 2 (a - /3) 2 (a - y) 2 (a - 5) 2 (p-/?) 2 (/? -«) 2 (/5-t) 2 (0-S) 2 (P-T) 2 (Y-«) 2 (Y-0) 2 (7-5) (p - 5) 2 (5 - a) 2 (5 - /3) 2 (5 - 7) 2 Interpret each form. (2). The equation of the ellipsoid may be written in the forms S 2 p/a - V 2 p/(3 = 1, where a is not parallel to ft T(p/y + Kpjb) = T(p/8 + tfp/7), rOup + pX)=x 2 -/* 2 . 134 VECTOR CALCULUS The planes a p cut the ellipsoid in circular sections on Tp = Tfi. These are the cyclic planes. Tfi is the mean semi-axis, Ufi the axis of the cylinder of revolution circumscribing the ellip- soid, a is normal to the plane of the ellipse of contact of the cylinder and the ellipsoid. In the second form let r 1 - -£, 7- 1 = - £> t 2 = n 2 - TJ, then the semi-axes are a=rX+7>, 6= ^~ TfX * > c=T\-T». T(\ - n) (3). The hyperboloid of two sheets is S 2 p/a + F 2 p//3 = 1. (4). The hyperboloid of one sheet is S 2 p/a + V 2 p/(3 = — 1. (5) . The elliptic paraboloid of revolution is SplP+V 2 p/(3 = 0. (6). The elliptic paraboloid is Sp/a + V 2 p/(3 = 0. (7). The hyperbolic paraboloid is Sp/a Sp/fi = Sp/y. (8). The torus is T(± bUarWap - p) = a, 2bTVap = ± (Tp 2 +b 2 - a 2 ), 4b 2 S 2 ap = 4b 2 T 2 p - (T 2 p + b 2 - a 2 ) 2 , Aa 2 T 2 p - 4b 2 S 2 ap = (T 2 p - b 2 + a 2 ) 2 , SU(p - «V (a 2 - b 2 ))l(p + cW (a 2 - b 2 )) = ± b/a, p = ± bJJoTWar + at/Y, r any vector. (9). Any surface is given by p = <p(u, v). APPLICATIONS 135 A developable is given by p = <p(t) + ucp'it). (10). A cone is f(U[p - a]) = 0. The quadric cone is SapSfip — p 2 = 0. The cone through a, (3, y, 8, e is S-V(Va(3V8e)V(V(3yVep)V(Vy8Vpa) = 0, which is Pascal's theorem on conies. The cones of revolution through X, n, v are The cones of revolution which touch S\p = 0, Sfxp = 0, Svp = 0, are The cone tangent to (p — a) 2 + c 2 = from /? is c 2 (p -a-$) 2 = V 2 (3(p - a). The polar plane of /3 is £/3(p — a) — — c 2 . The cone tangent to a p from 7 is (*i- F, S- 1 )( fl, J'- p i-0 -( S^S^-- SV?V?-- lY=0. \ a a a a / The cylinder with elements parallel to y is ( s *i- f 1- i )H- p ?) _(s>sl-sv>vl) 2 = o. \ a a a a) 136 VECTOR CALCULUS For further examples consult Joly : Manual of Quater- nions. 2. The Vector of Two Vectors Notations, If a and /3 are two fields, we shall call V-a(3 the torque of the two fields. Va(3 = Va(3 Hamilton, Tait, Joly, Heaviside, Foppl, Ferraris, Carvallo. cqS Grassman, Jahnke, Fehr. aX Gibbs, Wilson, Fischer, Jaumann, Jung. [a, /3] Lorentz, Gans, Bucherer, Abraham, Timer- ding. [a | /?] Caspary . a A j3 Burali-Forti, Marcolongo, Jung. aj8 Heun. Sin a/3 Macfarlane. Iaccb Peano. 1. Lines. The condition that two lines be parallel is that Vafi = 0. Therefore the equation of the line through the origin in the direction a is Vap — 0. The line through parallel to a is Va(p — fi) = or Vap = Va(3 = y. The perpendicular from 5 on the line Vap = 7 is — a~ l Vab + a~ l y. The line of intersection of the planes, S\p = a, S^p = b, is VpV\fx = a/x — 6X. If we have lines Vpa — y and Vp& = 8 then a vector from a point on the first to a point on the second is 5/3" 1 — 7a -1 + #/3 — ya. If now the lines in- tersect then we can choose x and y so that this vector will vanish, corresponding to the two coincident points, and thus S{bp~ l - ya~ l )$a = = S8a + Syp. APPLICATIONS 137 If we resolve the vector joining the two points parallel and perpendicular to Vaft we have* 5/3 -1 — ya~ l + xfi — ya = • (Va^S • VaP(bpr l - yoT 1 + zp - ya) = -(VaP)-\S5(x+ Spy) L a Fa/3 Fa/3 J L Va0 P Vap] - «-* f- SaPS ^~ + a 2 S JL 1 Vap Vap] Hence the vector perpendicular from the first line to the second is - (Vafl-KStct + Spy) and vectors to the intersections of this perpendicular with the first and second lines are respectively and ya x — a 1 \ 8 ' — ^— L Va(3 J * Note that (Va0)- l V(Vu0)(z0 - ya) = xp - ya (y« j S)- 1 F-7a/3(5 J 3- 1 - ya~ l ) = (Vc0)- l (- a'^Sfiya - (r l S<*&) Va ,(-^S^ + p-S^) 10 138 VECTOR CALCULUS The projections of the vectors a, y on any three rectangular axes give the Pluecker coordinates of the line. For applica- tions to linear complexes, etc., see Joly: Manual, p. 40, Guiot: Le Calcul Vectoriel et ses applications. 2. Congruence. The differential equation of a curve or set of curves forming a congruence whose tangents have given directions cr, that is, the vector lines of a vector field <r, is given by Vdpa = or its equivalent equation dp = adt. 3. Moment. The moment of the force /3 with respect to a point whose vector from an origin on the line of @ is a, is — Fa/3. If the point is the origin and the vector to some point in the line of application of the force is a, then the moment with respect to the origin is Vafi. If the point is on the line of application the moment obviously vanishes. If several forces have a common plane then the moments as to a point in the plane will have a common unit vector, the normal to the plane. If several forces are normal to the same plane, their points of application in the plane given by ft, ft, ft, • • • , their values being a\a> a 2 a, a s a, • • • , then the moments are F(aift + a 2 ft + a 3 ft + • • •)« [dyne cm.]. If we set «ift + 02ft + 03ft + • • • = ftai + a 2 + a z + • • •)/ then /3 is the vector to the mean point of application, which, in case the forces are the attractions of the earth upon a set of weighted points, is called the center of gravity. If ai + #2 + a 3 + • • • = 0, we cannot make this substitution. APPLICATIONS 139 4. Couple. A couple consists of two forces of equal magnitude, opposite directions and different lines of action. In such case the mean point becomes illusory and the sum of the moments for any point from which vectors to points on the lines of action of the forces are a h a 2 respectively, is V{a x - a 2 )P. But a\ — a 2 is a vector from one line of action to the other, and this sum of the moments is called the moment of the couple. It is evidently unchanged if the tensor of /? is increased and that of a\ — a 2 decreased in the same ratio, or vice versa. 5. Moment of Momentum. If the velocity of a moving mass m is a cm./sec, then the momentum of the mass is defined to be ma gr. cm./sec. The vector to the mass being p, the moment of momentum of the mass is defined to be Vpma = mVpa [gm. cm. 2 /sec.]. 6. Electric Intensity. If a medium is moving in a mag- netic field of density B gausses, with a velocity a cm./sec, then there will be set up in the medium an electromotive intensity E of value E=Fo-B-10~ 8 [volts/centimeter]. For any path the volts will be - fSd P E= + fSdpBa-10- 8 . If this be integrated around any complete circuit we shall arrive at the difference in electromotive force at the ends of the circuit. 7. Magnetic Intensity. If a magnetic medium is moving in an induction field of D lines, with a velocity a, then there will be produced in the medium at every point a magnetic 140 VECTOR CALCULUS intensity field H = OAwVDa [gilberts/cm.]. For any path the gilbertage will be OAirf SdpaD. 8. Moving Electric Field. If an electric field of induc- tion, of value D lines, is moving with a velocity a, then there will be produced in the medium at the point a mag- netic field of intensity H gilberts/cm. where H m OAirVaD. For a moving electron with charge e, this will be — (eUp/4:irTp 2 ). For a continuous stream of electrons along a path we would have the point being the origin. 9. Moving Magnetic Field. If a magnetic field of in- duction of value B gausses is moving with a velocity cr, it will produce at any given point in space an electric intensity E = V - BolO -8 volts per centimeter. 10. Torque. If a particle of length dp is in a field of intensity <r which tends to turn the particle along the lines of force, then the torque produced by the field upon the element is V-dpa. If a line runs from A to B, the total torque is // V-dpe. For instance if dp, or in case of a non-uniform distribution cdp, is the strength in magnetic units, maxwells, of a wire magnet from A to B, in a field a, then fIV-dpa or f/V-cdpa is the torque of the field upon the magnet. APPLICATIONS 141 11. Poynting Vector. An electric intensity E volts/cm. and magnetic intensity H gilberts/cm. at a point in space are accompanied by a flux of energy per cm. 2 R, given by the formula 4xR = — — [ergs/cm. 2 sec.]. This is the Poynting vector. 12. Force Density. The force density in dynes/cc. of a field of electric induction on a magnetic current is given by, F = 4ttFDG : 10 [dynes/cc], where D is the density in lines of electric displacement G is the magnetic current density in heavisides per cm. 2 . If the negative of F is considered we have the force per cc. required to hold a magnetic current in an electrostatic field of density D. The force density in dynes/cc. of a field of magnetic induction on a conductor carrying an electric current is F-ijr.il. A single moving charge e with velocity a will give F =AweVaiJiVaD. 13. Momentum of Field. The field momentum at a point where the electric induction is D lines and magnetic induction B gausses is T = 3-10 9 V- DB [gm. cm./sec.]. If the magnetic induction is due to a moving electric field then T = 0.047rF- D/jlVDct, and if the electric induction is due to a moving magnetic field, T = VB/cVaB. 47T-3-10 10 142 vector calculus 3. The Scalar of Three Vectors 1. Area and Pressure. If we consider two differential vectors from the point P, say dip, d 2 p, then the vector area of the parallelogram they form is Vdipd 2 p. If then we have a distribution of an areal character, such as pressure per square centimeter, /3, the pressure normal to the differential area will be in magnitude — S(3dipd 2 p. The vector Vdipd 2 p may be represented by dp or JJvdA. The vector pressure normal to the surface will be UpS(3dipd 2 p. There will also be a tangential pressure or shear, which is the other component of /3. 2. Flux. If j8 is any vector distribution the expression — S($d\pd 2 p is often called the flux of /? through the area Vd\pd 2 p. It is to be noted however that the dimensions of the result in physical units must be carefully considered. Thus the flux of magnetic intensity is of dimensions that do not correspond to any magnetic quantity. 3. Flow. If /3 is the velocity of a fluid in cm./sec, then the volume passing through the differential area per second is — Sfid\pd 2 p [cc./sec.]. 4. Energy Flux. The dimensions of the Poynting energy flow R show that it is the current of energy per second across a cm. 2 , hence the total flow per second through an area is -SRd^p-- 8 -™™^ [ergs/sec] In the case of a straight conductor carrying a current of electricity, we have at a distance a from the wire in a APPLICATIONS 143 direction at right angles to the wire directly away from it the value T R= (4ir)- 1 10 8 JS;(0.2Ja- 1 ). Consequently if we consider one centimeter of wire in length and the circumference of the circle of radius a we shall have a flux of energy for the centimeter equal to J(ft-«0 [jouks]. This is the usual J 2 R of a wire and is represented by heat. 5. Activity. For a moving conductor we have already expressed the vector E, and as the current density J can be computed from the intensity of the field (J = k E) we have then for the expression of the activity in watts per cubic centimeter of conductor A= -SaBhO- 8 = -S(V(rB)k(VaB)-lO- ie [watts]. Likewise in the case of the magnetomotive force due to motion and the magnetic current G = IH we have for the activity per cubic centimeter of circuit A=- SDaQ = - S-(VDa)l(VD(r)-10- 7 [watts]. 6. Volts. The total electromotive difference between two points in a conductor is the line-integral along the conductor - fSdpaBlCr* [volts]. 7. Gilberts. The total magnetomotive difference be- tween two points along a certain path is the line-integral — AirfSdpDo- [gilberts]. 4. Vector of Three Vectors 1. Stress. We find with no difficulty the equations V-a(Ua± Uy)y = ± TyTa(Ua± Uy) and V-a(Vay)y = — Say -V -ay. 144 VECTOR CALCULUS If now we have a state of stress in a medium, given by its three principal stresses in the form 0i = g — 7V dynes/cm. 2 normal to the plane orthogonal to U(U\+ Un), 92 = g — S\fx dynes/cm. 2 normal to the plane orthogonal to UV\fi, gz = g + T\p dynes/cm. 2 normal to the plane orthogonal to U{U\ - Un), gi < gi < gz, then the stress across the plane normal to /? is V\fo + 0. If the scalars g it g%, g z are dielectric constants in three directions (trirectangular) properly chosen, then the dis- placement is D = FXE/x + gE. If the scalars are magnetic permeability constants, B = V\Hfi + g W. If the scalars are coefficients of dilatation, then becomes (T-- VWp+gp. If the scalars are elasticity constants of the ether, then according to Fresnel's theory, the force on the ether is, for the ether displacement ft . . V\fo + gp. If the scalars are thermoelectric constants in a crystal, then D = FXQm + gQ. where Q is the flow of heat. If g = the scalars are TV, - TV, - SXfi. If V\fi = 0, the scalars are 7V> — TV> T\p, that is, practically — t along X and + t in all directions perpendicular to X. CHAPTER VIII DIFFERENTIALS AND INTEGRALS 1. DlFFEKENTIATION AS TO A SCALAR PARAMETER 1. Differential of p. If the vector p depends upon the scalar parameter t, say p = <p(t), then for two values of t which are supposed to be in the range of possible values for t Pi — Pi = <p(h) — <p(ti) t ti — t\ t% — ^1 If now we suppose that U < h < t 2 and that h and t 2 can independently approach the limit, t , then we shall call the limit of the fraction above, if there be such a limit, the right-hand derivative of p as to t, at t , and if t 2 < h < t , we shall call the limit the left-hand derivative of p as to t at t . In case these both exist and are equal, and if p has a value for t which is the limit of the two values of <p(ti), then we shall say that p is a continuous function of t at t and has a derivative as to t at to. There is no essential difference analytically between the function <p and the ordinary functions of a single real variable, and we will assume the ordinary theory as known. It is evident that for different values of t we may con- sider the locus of P which will be a continuous curve. Since p 2 — pi is a chord of the curve the limit above will give a vector along the tangent of the curve. Further the tensor of the derivative, Tp' = T(p'{t) y is the derivative of the length of the arc as to the parameter t. If the arc s is the parameter then the vector p' is a unit vector. 145 146 VECTOR CALCULUS EXAMPLES (1) The circle p = a cos -f sin 0, To: = Fft Sa0 = 0, p' = — a sin + cos 0. (2) The helix p = a cos + sin + 70, p' = — a sin + cos 0+7. (3) The conic _ a* 2 + 20< + 7 P at 2 +2U+C ' Multiplying out, t 2 (a - ap) + 2*(0 - bp) + (7 - cp) = for all values of £. For 2 = 0, p = 7/c, and for t = 00 , p = a/a, hence the curve goes through a/a and 7/c. We have rfp/d< = [t 2 (ba — a0) + <(ca — 07) + (c0 — by)] times scalar. Hence for t = 0, the direction of the tangent is 0/6 — 7/c at 7/c, for t = 00, the direction of the tangent is 0/6 — a/a at a/a. Since these vectors both run from the points of tangency to the point 0/6, the curve is a conic, tangent to the lines through 0/6 and the two points a/a and 7/c, at these two points. If the origin is taken at 0/6, so that p = w + 0/6, and if a! = a/a — 0/6, 7' = 7/c - 0/6, then at\a! - tt) - 26/tt + c(y f - w) = is the equation of the curve. If now we let w run along the diagonal of the parallelo- gram whose two sides are a'y' so that tt = x(a! + y'), then substituting we have at 2 x + 2btx - c(l - x) = 0, at 2 (l - x) - 2btx - ex = 0. DIFFERENTIALS 147 From these equations we have t 2 = c/a and x = Vac/2(Vac± b). These values of x give us the two points in which the diagonal in question cuts the curve. The middle point between these two is Referred to the original origin this gives for the center , ,, c a - 2b(3 + ay k= r + p b = — — =£— • 2(ac — b 2 ) If we calculate the point on the curve for bh + e ah+ b we shall find that for the points p 2 , pi we have J(p 2 + Pi) = k, so that k is the center of the curve and diametrically opposite points have parameters h and t 2 = — — r-x > ati ~t o an involutory substitution. If ac = b 2 , k becomes co ex- cept when also the numerator = 0. [Joly, Manual, Chap. VII, art. 48.] In general the equation of the tangent of any curve is IT = p + Xp'. We may also find the derivatives of functions of p, when p = (p(t), by substituting the value of p in the expression and differentiating as before. Thus let p = a cos 6 + P sin 6 where Ta # Tp. 148 VECTOR CALCULUS Then Tp = V [- a 2 cos 2 6 - 2Sap sin 6 cos 6 - 2 sin 2 6], We may then find the stationary values of Tp in the manner usual for any function. Thus differentiating after squaring a 2 sin 26 - 2Sa(3 cos 26 - fi 2 sin 26 = 0, tan 26 - 2Sap/(a 2 - /3 2 ). 2. Frenet-Serret Formulae. Since the arc is essentially the natural parameter of a curve we will suppose now that p is expressed in terms of s, and accents will mean only differentiation as to s. Then both p and p + dsp' are points upon the curve. The derivative of the latter gives p' + dsp", which is also a unit vector since the parameter is s. Thus the change in a unit vector along the tangent is dsp", and since this vector is a chord of a unit circle its limiting direction is perpendicular to p', and its quotient by ds has a length whose limit is the rate of change of the angle in the osculating plane of the tangent and a fixed direction in that plane which turns with the plane. That is to say, p" in direction is along the principal normal of the curve on the concave side, and in magnitude is the curmture of the curve, which we shall indicate by the notation Unit tangent is a = p', Unit normal is |9 = Up", curvature is Ci = Tp", Unit binormal is y = Va(3, so that Ciy = Vp'p". The rate of angular turn of the osculating plane per centi- meter of arc is found by differentiating the unit normal of the plane. Thus we have Ti = cf 2 hW - Fp'p"-c 2 ].' DIFFERENTIALS 149 But d 2 = T 2 p" = - Sp"p" and therefore Cl c 2 = - S P "p f ". Substituting for c 2 we have 71 - cr 3 [- Sp"p"Vp'p f " + SpV'Wl = cr z [Vp , Vp"Vp ,,, p"] = crWaVc 1 (3Vp'"c 1 p = crW-aPVp"^ = cr l VyVp'"p = cr l pSyp"' = - «lft where «i is written for the negative tensor of 71 and is the tortuosity. It may also be written in the form Again since /? = ya we have at once the relations j3i = 7i« + 7«i = «i7 ~~ C\a. Thus we have proved Frenet's formulae for any curve «i = erf, ft = ai7 — ci«, 71 = — a$. It is obvious now that we may express derivatives of any order in terms of a, ft y, and Oi, Ci, and the derivatives of ai and Ci. For instance we have Pi = OL, p 2 = fci, Pa = ftci + Pc 2 = fe + (701 — aci)ci, Pi = 0c 3 + 2{yai — aci)c 2 + (ya 2 — ac 2 )ci - ^( ai 2 + Cl 2 ) Cl . The vector w = aai + 7C1 is useful, for if 77 represents in turn each one of the vectors a, /3, 7, then 771 = Fa^ It is the vector along the rectifying line through the point. The centre of absolute curvature k is given by K = p - lip" m p + Pld. 150 VECTOR CALCULUS The centre of spherical curvature is given by a = k + yd/da • c{~ 1 = k — yc 2 /aiCi 2 . The polar line is the line through K in the direction of 7. It is the ultimate intersection of the normal planes. 3. Developable s. If we desire to study certain de- velopables belonging to the curve, a developable being the locus of intersections of a succession of planes, we proceed thus. The equation of a plane being S(w — p)rj = 0, where t is the vector to a variable point of the plane, and p is a point on the curve, while rj is any vector belonging to the curve, then the consecutive plane is S(t - p)f) + ds'd/dsS(w - p)r) = 0. The intersection of this and the preceding plane is the line whose equation is 7r = p + (— r)Sar) + t)lVr}r}i. This line lies wholly upon the developable. If we find a secOnd consecutive plane the intersection of all three is a point upon the cuspidal edge of the developable, which is also the locus of tangents of the cuspidal edge. This vector is tv = p + (VwySar} + 2Vr)7]iSar)i + Vr}7}iS^rjCi)/ST]r}ir]2' By substituting respectively for 77, a, ft 7, we arrive at the polar developable, the rectifying developable, the tangent- line developable. EXAMPLE Perform the substitutions mentioned. 4. Trajectories. If a curve be looked upon as the path of a moving point, that is, as a trajectory, then the param- eter becomes the time. In this case we find that (if p = dp/dt, etc.) the velocity is p = av, the acceleration is DIFFERENTIALS 151 p = ficiv 2 + av. The first term is the acceleration normal to the curve, the centrifugal force, the second term is the tangential acceleration. In case a particle is forced to describe a curve, the pressure upon the curve is given by (3civ 2 . There will be a second acceleration, p = a(v — wi 2 ) + (3(2cii + c 2 v) + yaiCiV. The last term represents a tendency per gram to draw the particle out of the osculating plane, that is, to rotate the plane of the orbit. 5. Expansion for p. If we take a point on the curve as origin, we may express p in the form p = sa + %cis 2 (3 — %s*(ci 2 a — c 2 /3 — cmy) — ^ 4 (3c 2 cia: ~~ £I C3 ~~ c * — Clttl2 l ~~ T[2c 2 ai + da 2 ]) EXERCISES 1. Every curve whose two curvatures are always in a constant ratio is a cylindrical helix. 2. The straight line is the only real curve of zero curvature every- where. 3. If the principal normals of a curve are everywhere parallel to a fixed plane it is a cylindrical helix. 4. The curve for which Ci = 1/ms, ai = 1/ns, is a helix on a circular cone, which cuts the elements of the cone under a constant angle. 5. The principal normal to a curve is normal to the locus of the centers of curvature at points where Ci is a maximum or minimum. 6. Show that if a curve lies upon a sphere, then cr 1 = A cos a + B sin a = C cos (a + e), A, B, C, e are constants. The converse is also true. 7. The binormals of a curve do not generate the tangent surface of a curve. 8. Find the conditions that the unit vectors of the moving trihedral afiy of a given curve remain at fixed angles to the unit vectors of the moving trihedral of another given curve. Two Parameters 6. Surfaces. If the variable vector p depends upon two arbitrary parameters it will terminate upon a surface of 152 VECTOR CALCULUS some kind. For instance if p = <p(u, v), then we may write for the total differential of p dp = dud/du(<p) -f- dvd/dv((p) = du<p u + dv<p v . We find then Fdp = £dw 2 + 2Fdudv + GW, where E = — ^ tt 2 , F = — S<p u <p v , G = — ^t, 2 . We have thus two differentials of p, one for » = constant, one for u = constant, which will be tangent to the para- metric curves upon the surface given by these equations, and may be designated by pidu, p 2 dv. The normal becomes then v = v Pl p 2 , Tv = V (EG - F 2 ) = H. For certain points or lines v may become indeterminate, the points or lines being then singular points or singular lines. 7. Curvatures. If we consider the point p and the point p + dupi -f- dvp 2 the two normals will be v and v + duV(p n p2 + P1P12) -f- dvV(pi2p2 + P1P22) + • • • which may be written v and v + dv. The equations of these lines are V(w - P )v =0, V(w- p- d P )(v + dv) = 0. They intersect if Sdpvdv = 0. Points for which this equation holds lie upon a line of DIFFERENTIALS 153 curvature so that this is the differential equation of such lines. If we expand the total differentials we have du 2 Spivi>i + 2dudvS(piw 2 + Pivv\) + dv 2 Sp 2 w 2 = 0. We may also write the equation in the form dp + xv + ydv = = pidu -\- p 2 dv + xv + yv\du + yv 2 dv. Multiply by (pi + yv\){p 2 + yv 2 ) and take the scalar part of the product, giving S(pi + yvi)(pi + P2#> = o = y 2 Svviv 2 + 2ySv{piv 2 + ^ip 2 ) + ^ 2 . The ultimate intersection of the two normals is given by t = p + dp + yv + y<&>, that is by yv. Hence we solve for yTv, giving two values R and R f which are the principal radii of curvature at the point. The product and the sum of the roots are re- spectively RR' = yy'Tv 2 - Tv%- Sw 1 v 2 ), R + R' = — 2TvSv(piv 2 -\- vip 2 )/Swiv 2 . The reciprocal of the first, and one-half the second divided by the first, that is, — Spvivt/v 4 and Sv(piv 2 + vip 2 )/Tv*, are the absolute curvature and the mean curvature of the surface at the point. The equation of the lines of curvature may be also written vSdpvdv = = V-VdpVvdv = VdpV(dv/vv) = VdpdUv. Hence the direction of dUv is that of a line of curvature, when du and dv are chosen so that dp follows the line of curvature. That is, along a line of curvature the change li 154 VECTOR CALCULUS in the direction of the unit normal is parallel to the line of curvature. When the mean curvature vanishes the surface is a minimal surface, the kind of surface that a soapfilm will take when it extends from one curve to another and the pressures on the two sides are equal. The pressure indeed is the product of the surface tension and twice the mean curvature, so that if the resultant pressure is zero, the mean curvature must vanish. If the radii are equal, as in a sphere, then the resultant pressure will be twice the surface tension divided by the radius, for each surface of the film, giving difference of pressure and air pressure = 4 times surface tension/radius. The difference of pressure is thus for a sphere of 4 cm. radius equal to the surface tension, that is, 27.45 dynes per cm. When a surface is developable the absolute curvature is zero, and conversely. Surfaces are said to have positive or negative curvature according as the absolute curvature is positive or negative. EXERCISES 1. The differential equation of spheres is Vp(p - a) = 0. 2. The differential equations of cylinders and cones are respectively Sva = 0, Sv(p - a) = 0. 3. The differential equation of a surface of revolution is Sapv = 0. 4. Why is the center of spherical curvature of a spherical curve not of necessity the center of the sphere? 5. Show how to find the vector to an umbilicus (the radii of curvature are equal at an umbilicus). 6. The differential equation of surfaces generated by lines that are perpendicular to the fixed line a is SVav<pVocv = 0, where <p is a linear function. DIFFERENTIALS 155 7. The differential equation of surfaces generated by lines that meet the fixed line V(p — (3) a = is SVvV{p - P)a<p{V V V(p - 0)a) = 0. 8. The differential equation of surfaces generated by equal and similarly situated ellipses is SV(Va&-p)v(VYa0-p) = 0. 9. Show that the catenoid p = xi + a cosh x/a(cos 8j + sin 6k) is a minimal surface, and that the two radii are db Tv, the normal which is drawn from the point to the axis. 2. Differentiation as to a Vector 1. Definition. Let q = /(p) be a function of p, either scalar, vector, or quaternion. Let p be changed to p + dt • a where a is a unit vector, then the change in q is given by dq= q' - q = f{p+ dt-a) - /(p), and dq/dt = Lim [/(p + dta) - f(p)]/dt as dt decreases. If we consider only the terms in first order of the infinitesimal scalar dt we can write dq = dtf(p, a) in which a will enter only linearly. In a linear function of a however we can introduce the multiplier into every term in a and write dta = dp, so that we have dq a linear function of dp, dq = f'(p, dp). It needs to be noted that the vector a is a function of the variable dt, although a unit vector. The differential of q is of course a function of the direction of dp in general, but the direction may be arbitrary, or be a function of the variable vector p. It may very well happen that the limit obtained above may be different for a given function / according to the direction of the vector a. In general, we intend to consider the 156 VECTOR CALCULUS vector dp as having a purely arbitrary direction unless the contrary is stated. EXAMPLES (1) Let q = " P 2 . Then dq = - [p2 + 2dtS-pa - p 2 ] = - 2dtSpa = - 2Spdp. Also since q = T 2 p we have dq = 2TpdTp=- 2Spdp, whence dTp/Tp = Sdp/p, or dTp = - SUpdp. (2) From the definition we have d(qr) = dq-r + g-dr, hence d(Tp-Up) = dTp-Up+ Tp-dUp = dp and utilizing the result of the preceding example, we have dUp/Up = Vdplp. Also we may write dUp = — Vdpp-p/T 3 p = pVdpp/T 3 p = p~ l VpdpjTp, etc. This equation asserts that the dif- ' ferential of Up is the part of the arbitrary differential of p perpendicular to Up, divided by the length of p, that is, it is the differential angle of the two directions of p laid off in the direction perpendicular to p in the plane of p and dp. In case dp is along the direction of p itself, dUp = 0. (3) We have since d{pp~ l p) = dp = dp-p~ l p + pd{p~ l )p + pp~ l dp = 2dp + pd(p' l )p 9 DIFFERENTIALS 157 and thence dp = — pd{p~ l )p, i.p-i = - p-Hpp- 1 = [p-'Spdp - p-WpdpWFp = p-'dp-p/rp. That is, the differential of p~ l is the image of dp in p divided by the square of Tp. Hence diVap)- 1 = (Vap)- l Vadp-VapjTWap. This vanishes if dp is parallel to a. (4) If x = — a 2 /p then dir = — a 2 p~ l dppj T 2 p, and for two different values of dp, as dip, dip, we have diir/diTT = p~ l dipld\p-p. Therefore in the process of "inverting" or taking the "electrical image " we find that the biradial of two dif- ferential vectors is merely reflected in p. Interpret this. (5) T- = c is a family of spheres with a and — a as p — a limit points. For a differential dp confined to the surface of any sphere we have then Sdp[(p + a)- 1 - (p - a)- 1 ] = 0. A plane section through a can be written Syap = 0, in which Syadp = gives a differential confined to the plane. Therefore a differential tangent to the line of intersection of any plane and any sphere will satisfy the equation Vdp[VVya«p + a)- 1 - (p - a)" 1 )] - 0. But the expression in the () is a tangent line to any sphere which passes through A and — A. For the equation of such a sphere would be p 2 - 2Sadp - a 2 = 158 VECTOR CALCULUS where 5 is any vector, hence for any dp along the sphere, S(p — VaS)dp = 0. But (p + a) -1 — (p — a) -1 is parallel to a(p 2 + « 2 ) - 2pSap and 5(p - Va8)[a(p 2 + a 2 ) — 2piSap] = — Sap[p 2 — a 2 — 2£pa5]. For points on the sphere the [] vanishes, hence the vector in question is a tangent line. Also Vttt is perpendicular to it or r, therefore the differential equation above shows that the tangent dp of the intersection of the plane and the sphere of the system is perpendicular to a sphere through A and —A. Hence all spheres of the set cut orthogonally any sphere through A and —A. (6) The equation SU = e is a familv of tores pro- p—a duced by the rotation of a system of circles about their radical axis. From this we have SU(p + a)(p-a) = -e, VU(p + a)(p - a) - V (1 - e 2 )UVap = a. Differentiating the scalar equation we have L P+ OL + TJ(p + a)V-^- >U(p - a)l = P — OL J or Sadp[(p + a)- 1 - (p - a)- 1 ] = 0. Now in a meridian section a is constant so that Vdp[(p + a)" 1 - (p - a)" 1 ] - and dp is for such section tangent to a sphere through A and —A. EXERCISES 1. The potential due to a mass m at the distance Tp is m/Tp in DIFFERENTIALS 159 gravitation units. Find the differential of the potential in any direc- tion, and determine in what directions it is zero. 2. The magnetic force at the point P due to an infinite straight wire carrying a current a is H = — 2h/Vap. Find the differential of this and determine in what direction, if any, it is zero. For Vdpa = 0, dH = 0; for dp = dsVa^Vap, dH = - Hds/TV<r P ; for dp = dsUVap, dH - V<rUd8./TV<rp. 3. The potential of a small magnet a at the origin on a particle of free magnetism at p is u = Sap/T 3 p. Find the variation in directions Up, UVap, UaVap. 4. The attraction of gravitation at a point P per unit mass in gravita- tion units is a = - Up/T*p. Find the differential of <x in the directions Up and F/3p. da = - {pHp - SpSpdp)/T 5 P ; parallel to p, - 2/p 3 ; perpendicular, UV@p/T s p. 5. The force exerted upon a particle of magnetism at p by an element of current a at the origin is H = - V<x P IT s p. Then dH = {pWadp - 3Va P Spdp)/T 5 P ; in the direction of p, 37a/p 3 ; in the direction Vap, — VaUVap/T 3 p. 6. The vector force exerted by an infinitesimal plane current at the origin perpendicular to a, upon a magnetic particle or pole at p is a = (ap 2 - SpSap)/T* P . Find its variation in various directions. 2. Differential of Quaternion. We may define differen- tials of functions of quaternions in the same manner as functions of vectors. Thus we have T 2 q — qKq so that 2TqdTq = d(qKq) = [(q + dtUq)(Kq + dtUKq) - qKq] = dtlqUKq + UqKq] = qKdq -f- dqKq = ZSqKdq = 2SdqKq. That is, dTq = SdqUKq = SdqUq' 1 = TqSdq/q or dTq/Tq = Sdq/q. 160 VECTOR CALCULUS In the same manner we prove the other following formulae. dUq/Uq = Vdq/q, dSq = Sdq, dVq = Vdq, dKq = Kdq, S(dUq)/Uq = 0, dSUq = SUqV(dq/q) = - S(dq/qUVq')TVUq = TVUqdzq, dVUq = VUKqV(dqlq), dTVUq = - SdUqUVq = SUqdzq, d-q 2 = 2Sqdq + 2Sq- Vdq + 2Sdq- Vq, d-q~ l = — q~ l dqq~ l , d-qaq- 1 = - 2V -qdq^qVaq- 1 = 2V-dq(Va)q-\ that is, if r = gag -1 , then dr = 2V(dqjq>r) = - 2V(q-dq- l -r) = 2V(Vdq/q)r - 2q-V 'V{q- l dq-a)q~ l dUVq= V'Vdq/Vq-UVq, dzq= S[dqKUVq-q)]. We define when 7a = 1 a x = cos • irx/2 + sin • 7nc/2 • a = catf • %tx; thus d-a* = tt/2-o:^ 1 ^. If Ta # 1, then d-a x = dz[log 7W* + tt/2 -a x+1 /Ta\, 3. Extremals. For a stationary value of /(p) in the vicinity of a point p we have ay(p) = 0. If /(p) is to be stationary and at the same time the terminal point of p is to remain on some surface, or in general if p is to be subject *Tait, Quaternions, 3d ed., p. 97. DIFFERENTIALS 161 to certain conditioning equations, we must also have, if there is one equation, q(p) = 0, dq(p) = 0, and if there are two equations, g(p) = and h(p) = 0, then also dg(p) = 0, dh(p) = 0. Whether in all these different cases /(p) attains a maximum of numerical value or a minimum, or otherwise, we will consider later. EXERCISES 1. g( p ) = (p — a) 2 -f- « 2 = 0, find stationary values of Tp = f(j>). Differentiating both expressions, Sdp(p —a) = = Sdpp, for all values of dp. Hence we must have dp parallel to V • tp where t is arbitrary, and hence Srp(p — a) = 0, for all values of r. Therefore we must have Vp(p — a) = 0, or Yap = 0, or p = ya. Substituting and solving for y, y = 1 ± a/Tcc, p = a ± aUa. . 2. g( p ) = ( p — «) 2 + a 2 = 0. Find stationary values of &/3p. Sdp(p - a) = = *S/3ap, whence dpP.WjS, £'T,3(p - a) = 0, 7/3(p - a) = 0. p - a =y0, y = a/T(3, p = a ± at//3. 3. ^( p ) = (p — a )2 -f a 2 = 0, &G>) = *S/3p = 0, find stationary values of Tp. Sdp( P - a) = = Spdp = Spdp, whence S-p0(p — a) « - £pa/S, and since £p0 = 0, p = yV-fiVafi. p = V0VaP(l ± V[a 2 - S*a0)/TVal3). 4. #(p) = p 2 — SapSpp + a 2 = 0. Find stationary values of Tp. £d P p = o = £dp(p - «S/8p - jSflap), p = x(aS$p + /8/Sap) = (a£/3p + pSap)/(Sa(3 ± Ta0), whence Sap = TaSpU/3, = SpU(3(Ua ± U0)/(SUaU0 T 1). Substituting in the first equation, we find SpUp, thence p. 5. Sfip = c, >Sap = c', find stationary values of Tp. Sd P p = Sadp = Spdp = 0, p = xa + y$ and z£a/3 -f 2//3 2 = c, xa 2 + ySafi = c', whence x and y. 162 VECTOR CALCULUS 6. Find stationary values of Sap when (p — a) 2 -f- a 2 = 0. Sctdp = = Sdp(p - a); hence p = ya = a ± aJ7a and Sap = a 2 ± aTa. 7. Find stationary values for Sap when p 2 — SppSyp + a 2 = 0. Sadp = = £d P (p - )857P - ySfip), P =xa+ fiSyp + ySfip, etc. 8. Find stationary values of TV8p when (p - a) 2 + a 2 = 0. 9. Find stationary values of SaUp when (p - a) 2 + a 2 = 0. 10. Find stationary values of SaUpSpUp when Syp + c = 0. 4. Nabla. The rate of variation in a given direction of a function of p is found by taking dp in the given direction. Since df(p) is linear in dp it may always be written in the form where $ is a linear quaternion, vector, or scalar function of dp. In case / is a scalar function, $ takes the form — Sdpv, where v is a function of p, which is usually independent of dp. In case v is independent of the direction of dp, we call / a continuous, generally differentiable, function. Functions may be easily constructed for which v varies with the direction of dp. If when v is independent of dp we take differentials in three directions which are not in the same plane, we have pS - dipd 2 pd 3 p = V'dipd 2 p-Sd 3 pp + V '• d 2 pd 3 p • Sdipp + V - d 3 pdipSd 2 pp = — V 'd 1 pd 2 p'd 3 f '— Vd 2 pd 3 p-dif — V-d 3 pdip-d 2 f. DIFFERENTIALS 163 It is evident that if we divide through by Sdipdipdzp, the different terms will be differential coefficients. The entire expression may be looked upon as a differential operation upon/, which we will designate by V. Thus we have v= V/ = _ ( Vdipdip - d z + V- d 2 pd s p • di + V- d^pdip ■ d 2 ) ,, , S • dipdipdzp We may then write df( P ) = - SdpVfip). If the three differentials are in three mutually rectangular directions, say i, j, k, then V = id/dx + jd/dy + kd/dz. It is easy to find V/ for any scalar function which is gener- ally differentiate from the equation for df(p) above, that is, df(p) = — SdpVf. For instance, VSap = - a, Vp 2 = - 2p, VTp = Up, V(Tp) n = nTp n - l Up = nTp n ~ 2 -p, V TVap = TJVap-a, VSaUp = - p-WUpa, V • SapSpp = - pSd$ - Vap(3, V-log TVap = -^~, vap VT(p - a)-' = - U(p - a)lT\p- a), VSaUpS(3Up = p-WpVap^P, Vlog Tp= U P /Tp= -p~\ 1 V(ZpA*) = - p~ 1 UVpa = pUVap 5. Gradient. If we consider the level surfaces of /(p), /(p) = C, then we have generally for dp on such surface or tangent to it S dp p = = df(p) where p. is the normal of the 164 VECTOR CALCULUS surface. Since Sdp\7f — and since the two expressions hold for all values of dp in a plane M = *V/, or since the tensor of p. is arbitrary, we may say V/(p) is the normal to the level surface of /(p) at p. It is called the gradient of /(p), and by many authors, particularly in books on electricity and magnetism, is written grad. p. The gradient is sometimes defined to be only the tensor of V/, and sometimes is taken as — V/. Care must be exercised to ascertain the usage of each author. Since the rate of change of /(p) in the direction a is — &*V/(p), it follows that the rate is a maximum for the direction that coincides with UVf, hence the gradient V/(p) gives the maximum rate of change off(p) in direction and size. That is, TVf is the maximum rate of change of /(p) and UVfis the direction in which the point P must be moved in order that /(p) shall have its maximum rate of change. 6. Nabla Products. The operator V is sometimes called the Hamiltonian and it may be applied to vectors as well as to scalars, yielding very important expressions. These we shall have occasion to study at length farther on. It will be sufficient here to notice the effect of applying V and its combinations to various expressions. It is to be ob- served that VQ may be found from dq, by writing dq = $-dp, then VQ = i$i + j$j + k$k. For examples we have Vp = {Vdipdzp-dzp + Vd 2 pdsp-dip + Vdzpdip • d 2 p) I '(— Sdipd 2 pd 3 p) = - 3 since the vector part of the expression vanishes. DIFFERENTIALS 165 Vp _1 = — (Vdipd 2 p-p~ 1 d 3 pp~ 1 + •••)/(— Sd 1 pd 2 pd 3 p) Since - - P" 2 . dUp = V^ • Up, dTp = - SUpdp. Hence VUp = 2iV--Up= -~, VTp= Up. p Tp Expressions of the form 2F(i, i, Q) are often written F($ > r> Q)> a notation due to McAulay. Vap = a, VfaSfap + cx 2 S/3 2 p + mSfop) = - 0m + /5 2 a 2 + 1830:3), VFap = 2a, VVap(3 = &xft VSapVfip = - Sapp + 3/3£a<p - pSa(3, VVaUp= (a + p^Sap)/ Tp, V • TTap = C/Fap • a, VTVpVap - (Fap + ap)UVpVap, VVap/T 3 p m (ap 2 - SpSap)/T 5 p, V ' UV « P =Tkp> VUVpVap = ^P-, V(Vap)-i=0, V (g)=0. EXERCISE Show that (Fa/3 -<l>y -+- y0y<£>a + Vy<x'3?P)/Sa0y is independent of a, /3, 7, where $ is any rational linear function (scalar, vector, or quaternion) of the vector following it. If <*> = S8( ) + 2ai<S/3i( ) the ex- pression is 5 + S/Siai. Notation for Derivatives of Vectors Directional derivative - SaV, Tait, Joly. a- V, Gibbs, Wilson, Jaumann, Jung. Tp -a, Burali-Forti, Marcolongo. 166 VECTOR CALCULUS Circuital derivative VaV, Tait, Joly. a X V, Gibbs, Wilson, Jaumann, Jung. Projection of directional derivative on the direction. S-<r l vSau, Tait, Joly. — > Fischer. da Projection of directional derivative perpendicular to the direction V-trhi'SV'a, Tait, Joly. —— * Fischer. da Gradient of a scalar V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo, Bucherer. grad, Lorentz, Gans, Abraham, Burali-Forti, Marcolongo, Peano, Jaumann, Jung. . — grad, Jahnke, Fehr. [Fischer's multiplication follows Gibbs, d/dr d p. , being after the operand, the whole being dr read from right to left; e.g., Fischer's Vfl is equiv. to — vSV.] Gradient of a vector V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo. grad, Jaumann, Jung. -=- > Fischer. dr 7. Directional Derivative. One of the most important operators in which V occurs is— SaV, which gives, the DIFFERENTIALS 167 rate of variation of a function in the direction of the unit vector a. The operation is called directional differentiating. SaV'Sfo = - SaP, SaV-p 2 = - 2Sap, SaVTp - SaUp, SaVTp- 1 = - Sap/Tp* = UY^p- 2 , SaVTVap= 0, SaV-Up= - ^~ • An iteration of this operator upon Tp~ l gives the series of rational spherical and solid harmonics as follows : - SaVTp- 1 = - Sap/Tp* = UYiTff*, Sl3VSaVTp- 1 = (3SapS(3p+ Tp 2 Sa(3)Tp- 5 = 2\Y 2 Tp~\ SyVSWSaVTp- 1 = - (3.5SapS(3pSyp + 32S(3ySapTp 2 )Tp- 7 = 3\Y 3 Tp~\ For an n axial harmonic we apply n operators, giving Y n = S.(- l) 8 (2n - 2s)!/[2 n -*nl(n - s)l\ES n - 28 aUpS s a 1 a 2 , ^ s^ n/2. The summation runs over n — 2s factors of the type SaiUpSoi2Up • - • and s factors of the type SajCtjSotnar - - , each subscript occurring but once in a given term. The expressions Y are the surface harmonics, and the expressions arising from the differentiation are the solid harmonics of negative order. When multiplied by Tp 2n+1 we have corresponding solid harmonics of positive order. The use of harmonics will be considered later. 8. Circuital Derivative. Another important operator is Va\7 called the circuital derivative. It gives the areal density of the circulation, that is to say, if we integrate the function combined with dp in any linear way, around an infinitesimal loop, the limit of the ratio of this to the area of the loop is the circuital derivative, a being the normal to the area. We give a few of its formulae. We may also 168 VECTOR CALCULUS find it from the differential, for if dQ = $dp, Fa V • Q VaV • Tp - VaUp, FaV • Tp n = nTp n ~ 2 Vap t VaV - Up = (Sap 2 - pSap)/Tp\ VaV-SQp = F/3a, Fa V • V(3p = a(3+ S-aP, FaV -ft> = 2Sa(3, FaV • 7Tft> - - V-apUVpp, FaV -p - - 2a, Fa V • (aiSftp + a 2 »S/3 2 p + a 3 S/3 3 p) - Sa(« A + "A + a 3 fo) + FaiFa/3i + Fa 2 Fa/3 2 + Fa 3 Fa/3 3 . 9. Solutions of VQ = 0, V 2 Q =0. In a preceding formula we saw that V(Vap)~ l = 0. We can easily find a number of such vectors, for if we apply Sa V to any vector of this kind we shall arrive at a new vector of the same kind. The two operators V and Sa V • are commutative in their operation. For instance we have d(Vap)~ l = - (VapyWadp-iVap)- 1 ; hence T = ^V-(Fap)- 1 = {Vap)- l V$a>{Vap)- 1 is a new vector which gives Vr = 0. The series can easily be extended indefinitely. Another series is the one de- rived from Up/T 2 p. This vector is equal to p/T 3 p, and its differential is (-p 2 dp+SSdpp.p)/T% The new vector for which the gradient vanishes is then (-ap 2 +3Sap-p)/7V The latter case however is easily seen to arise from the vector V Tp~ l , and hence is the first step in the process of using V twice, and it is evident that S7 2 Tp~ l = 0. So also the first case above is the first step in applying V 2 to log TVap-a~ l so that V 2 (log TVap-a) = 0. Functions of p that satisfy this partial differential equation are called DIFFERENTIALS 169 harmonic functions. That is,/(p) is harmonic if V 2 /(p) = 0. Indeed if we start with any harmonic scalar function of p and apply V we shall have a vector whose gradient van- ishes, and it will be the beginning of a series of such vectors produced by applying &*iV, Sa 2 V, • • • to it. However we may also apply the same operators to the original harmonic function deriving a series of harmonics. From these can be produced a series of vectors of the type in question. V 2 • F(p) is called the concentration of F(p) . The concentra- tion vanishes for a harmonic function. EXERCISES Show that the following are harmonic functions of p: 1. Tp- 1 tan" 1 Sap/Spp, where a and /? are perpendicular unit vectors, 2. Tf* log tan ^ Z £ 3. where and tan -1 Sap/S/3p Sa(3 = a 2 = £2 = _ 1# 4. logtan^ z - • £j CL 10. Harmonics. We may note that if u, v are two scalar functions of p, then V -uv = u Vfl + v\7u and thus V 2 -uv = u\7h + vV 2 u + 2SVuVv. Hence the product of two harmonics is not necessarily harmonic, unless the gradient of each is perpendicular to the gradient of the other. Also if u is harmonic, then \7 2 -uv = u\7 2 v + 2SVu\7v. 12 170 VECTOR CALCULUS If u is harmonic and of degree n homogeneously in p, then w/7p 2n+I is a harmonic of degree — (n + 1). For V 2 (fp 2n+1)-1 . V[ _ ( 2n+ l) p r p -2n-3] = - (2n+ l)(2n)Tp~ 2n - 3 and SVuVTp- 2 "- 1 = - (2n+ l)Tp- 2n -*SVup = (2n+ l)(2n)uTp- 2n -*; hence V 2 -u/Tp 2n+1 - 0. In this case w is a solid harmonic of degree n and uTp~ 2n ~ l is a solid harmonic of degree — n — 1. Also uTp" 11 is a corresponding surface harmonic. The converse is true. EXAMPLES OF HARMONICS Degree n = 0; <p = tan -1 — - £>pp where Sc& = 0, a 2 = /3 2 = - 1; ^ = log cot ^/ -a 2 = - 1; a S-a(3UpSapS(3p/V 2 -a(3p; Sa(3UpS(a + 0)pS(a - /3)p/F 2 a/3p. The gradients of these as well as the result of any opera- tion Sy V are solid harmonics of degree — 1, hence multiply- ing the results by Tp[n = 1, 2n — 1 = 1] gives harmonics again of degree 0. These will be, of course, rational harmonics but not integral. Taking the gradient again or operating by $71 V any number of times will give harmonics of higher negative degree. Multiplying any one of degree — n by Tp 2n ~ 1 will give a solid harmonic of degree n — 1. Degree n = — 1. Any harmonic of degree divided by Tp, for example, 1/Tp, ip/Tp, f/Tp, Saf3UpSaUpS(3p/V 2 a(3p, • • • , DIFFERENTIALS 171 Degree n = — 2. SaUp/p 2 , <pSa(3Up/p 2 , xPSa(3Up/p 2 + P" 2 • • • . Degree n = 1. Sop, *>&*ft>, ^Softa + 7p • • • . Other degrees may easily be found. 11. Rational Integral Harmonics. The most interesting harmonics from the point of view of application are the rational integral harmonics. For a given degree n there are 2n + 1 independent rational integral harmonics. If these are divided by Tp n we have the spherical harmonics of order n. When these are set equal to a constant the level surfaces will be cones and the intersections of these with a unit sphere give the lines of level of the spherical harmonics of the given order. A list of these follow for certain orders. Drawings are found in Maxwell's Electricity and Magnetism. Rational integral harmonics, Degree 1. Sap, S(3p, Syp, a, ft, y a trirectangular unit system. Degree 2. SapS(3p, SfoSyp, SypSap, 3S 2 ap + p 2 , S 2 ap - s 2 p P . These correspond to the operators 7p 5 [£ 2 7V, SyVSaV, SyVSPV, S(a + 0) VS(a - 0) V, SaVSQV] on Tp'K Degree 3. Representing Sap by — x, Sfip by — y, Syp by — z, SaV by — D x , S(3V by — D y , SyV by — D z we have 2z 3 — 3x 2 z — 3y 2 z, 4:Z 2 x — x* — y 2 x, A.z 2 y — x 2 y — y 3 , x 2 z — y 2 z, xyz, x z — 3xy 2 , 3x 2 y — y 3 corresponding to 7)3 7)3 7)3 7) 3 _ 7) 3 7) 3 _ Q7) 3 ^ zzz ) -lszzx , Lf zzy , ^xxz > J^xyz , ^xxx > OU X yy , 7) 3 _ Q7) 3 ■LSyyy j OJ^xxy • 172 VECTOR CALCULUS Degree 4. 3z 4 + 3y 4 + 8z 4 + 6*y - 24z 2 z 2 - 24yV, *z(4z 2 - Sx 2 - 3y 2 ), yz(4z 2 - 3^ - 3i/ 2 ) (^ _ y 2 )(6z 2 — x 2 — y 2 ), xy(6z 2 — x 2 — y 2 ), xz(x> - Sy 2 ), yzQx 2 - y 2 ), x* + y* - My 2 , xyix 2 - y 2 ) 7) 4 7) 4 7) 4 is zzzz ) *-* zzzx y -L/zzzy D 4 - ■LS ZZXX . T) 4 7) 4 7) 4 _ OT) 4 *s zzyy j M* zzxy i J^xxxz *->±s X yyz j 7) 4 1J yyyz _ Q7) 4 7) 417) 4_ ft/) 4 oiyxxyz ) Uxxxx T ^ yyyy ^^xxyy y D 4 — 7) 4 J^xxxy ^xyyy • The curves of the intersections of these cones with the unit sphere are inside of zero-lines as follows : Degree 1. Equator, standard meridian, longitude 90°. Degree 2. Latitudes ± sin -1 JV 3, equator and standard meridian, equator and longitude 90°, longitude ± 45°. Standard meridian and 90°. Degree 3. Latitudes 0°, db sin -1 V 0.6, latitudes ± sin -1 V 0.2 and standard meridian, latitudes ± sin -1 V 0.2 and 90° longitude, equator, longitude ± 45°, equator, longitudes 0°, 90°. Longitudes ± 30°, 90°, longitudes ± 60°, 0°. 12. Variable System of Trirectangular Unit Vectors. We will consider next a field which contains at every point a system of three lines which are mutually perpendicular. That is, the lines in one direction are given by a, say, at the same point another set by ft and the third set by y. Each is a given function of p, subject to the conditions a/3 = 7, /?7 = a, ya = /?, a 2 = (3 2 = y 2 = — 1. For example, in the ordinary congruence, a being the unit tangent at any point of one line of the congruence, then the normal and the binormal are determined and would be ]S and 7. However /? and 7 may be other perpendicular DIFFERENTIALS 173 lines in the plane normal to a. If we follow the vector line for /3 after we leave the point we shall get a determinate curve, provided we consider a to be its normal. We may however draw any surface through the point which has a for its normal and then on the surface draw any curve through the point. All such curves can serve as ft curves but a might not be their principal normal. It can happen therefore that the j8 curves and the y curves may start out from the point on different surfaces. However a, (3, and y are definite functions of the position of the point P, with the condition that they are unit vectors and mutually perpendicular. If we go to a new position infinitesimally close, a becomes a + da, ft becomes fi + dp, and y becomes y + dy. The new vectors are unit vectors and mutually perpendicular, hence we have at once S-ada = S-pdp = S>ydy = 0, Sadp = - S(3da, n . Spdy = - Sydp, Syda = - Sady. {L) These equations are used frequently in making reductions. We have likewise since a 2 = — 1, Va-a = - VW, V/3-/3 = - VW, (2) vy-t = - v'rr'j where the accent on the V indicates that it operates only on the accented symbols following. Similarly we have Va-j8 + V(3-a= - V'a0' - V'j&x', etc. (3) We notice also that S-a(SQV)a = 0, S-a(SQV)0 = - S-p(S()V)a, etc. (4) We now operate on the equation y = afi with V, and 174 VECTOR CALCULUS remember that for any two vectors X/x we have X/x = — juX + 2<SX/x, whence V7 = Va-j3 + V'aP' = Va-/3 - V/3-a + 2V'Sa(3'. (5) The corresponding equations for the other two vectors are found by changing the letters cyclically. Multiply every term into y and we have Vt-7 = Vo-a + Vj8-|8 + 2V'Sct(3'-y. (G) If now we take the scalar of both sides we have SyVy = SaVa + Sj8V0 + 2SyV'Sa(3'. (7) We set now 2p = + &*Va + SjSVjS + #7 Vt (8) and the equation (7) gives, with the similar equations deduced by cyclic interchange of the letters, SyVSctP - - SyV'Sa'Q = - p + S7V7, SaV'SPy' m - SaV'Sfi'y = - P + 5a V«, SpVSya' = - SpV'Sy'a = - p + S0V/3, - S-Tf- 5a V • y] = 5a V • £77' = |&* V ■ 7 2 = 0, ( j - 5-a[- 5aV-7] = - SaV-Sa'y = Sy(— SaV -a) = Sy(u(3 + vy) = — v. That is to say, the rate of change of y, if the point is moved along a, is ]8(5aVa — p). Likewise dfi/ds = — 7(— p + 5aVa)— -ya. The trihedral therefore rotates about a with the rate (p — SaVa) as its vertex moves along a. Now we let t a = + p - SaVa. (10) We may also write at once, similarly, h - + V - S0VA * 7 = + p - 5 7 V7, (10) from which we derive t + V+ <»-+■* (ID DIFFERENTIALS 175 It is also evident that *. + U = SrVy, t fi + *, - SaVa, / 7 + / a = 5/3 V/3. (12) The expressions on the left hold good for any two per- pendicular unit vectors in the plane normal to the vector on the right, and hence if we divide each by 2 and call the result the mean rotatory deviation for the trajectories of the vector on the right, we have TjSctVct = mean rotatory deviation for a. Again the negative rotation for the trajectory gives what we have called previously the rotatory deviation of a along j3. Hence, as a similar statement holds for y, the mean rotatory deviation is one half the sum of the rotatory deviations. Hence %Sa\7a is the negative rate of rotation of the section of a tube of infinitesimal size, whose central trajectory is a, about a, as the point moves along a. Or we may go back to (9) and see that SaVa = (+ p ~ SPVB) + (+ V ~ SyVy) = - SpV'Sya' + SyV'Sfa', which gives the rotatory deviations directly. The scalar of (5) and the like equations are SVa = SyVP - Sj3\7y, SVP = SaVy - SyVa, (13) SVy = SfiVa - SaVP, We multiply next (5) by a and take the scalar, giving SyVa = - SaV'Sfia' = SaV'Sa(3 f , SfiVa m - SaVSay* = SaV'Sya', SaVP = - SpV'Sy? = St3V'S(3y', £ T V/3 = - SpV'Spa' = S(3V'Sa(3', SfiVy = - SyV'Say' = SyV'Sya', SaVy = - SyV'Sy(3' m SyV'Sfiy'. (14) 176 VECTOR CALCULUS We can therefore write SVa = - SWSPa' - SyV'Sya', that is SVa equals the negative sum of the projection of the rate of change of a along (3 on /3, and the rate of change of a along y on y. But these are the divergent deviations of a and hence — SVa is the geometric divergence of the section. It gives the rate of the expansion of the area of the cross-section of the tube around a. We may write the corresponding equations of /8 and y. Again we have FVa = — aSaVa — (3S(3Va — ySyVoc = cx(t a - v) - PSy(SaV-a) + ySp(SaV-a) = a(t a — p) — Va(SaV-a). Now from the Frenet formulae — Sa'V 'Ol = c a v, where c a is the curvature of the trajectory and v is the principal normal. Hence Wa = a(t a - p) + CJh (15) where /i is the binormal of the trajectory. We find there- fore that VVd consists of the sum of two vectors of which one is twice the rate of rotation of the section or an elemen- tary cube about a, measured along a, and the other is twice the rate of rotation of the elementary cube about the binormal measured along the binormal.* But we will see * This should not be confused with the rotation of a rigid area mov- ing along a curve. The infinitesimal area changes its shape since each point of it has the same velocity. As a deformable area it rotates (i.e. the invariant line of the deformation) with half the curvature as its rate. The student should picture a circle as becoming an ellipse, which ellipse also rotates about its center. DIFFERENTIALS 177 later that this sum is the vector which represents twice the rate of rotation of the cube and the axis as it moves along the trajectory of a. Hence this is what we have called the geometric curl. We may now consider any vector a defining a vector field not usually a unit vector. Since a = TaUa, we have SVa = SUaVTa + TaSvUa. The last term is the geometric convergence multiplied by the length of a, that is, it is the convergence of a section at the end of a. The first term is the negative rate of change of TV along a. The two together give therefore the rate of decrease of an infinitesimal volume cut off from the vector tube, as it moves along the tube. In the lan- guage of physics, this is the convergence of a. Similarly we have Wa= VvTaU<r+ TaWUa. The last term is the double rate of rotation of an elementary cube at the end of a, while the first term is a rotation about that part of the gradient of Ta which is perpendicular to Ua. It is, indeed, for a small elementary cube a shear of one of the faces perpendicular to Ua, which gives, as we have seen, twice the rate of rotation corresponding. Con- sequently VVa is twice the vector rotation of the elemen- tary cube. EXAMPLES (1) Show that aSVa + (3S V0 + yS Vt = - VaWa - V(3WP - VyV\7y. (2) Show that if dipt) is zero VaWa = 0. This is the condition that the lines of the congruence be straight. It is necessary and sufficient. 178 VECTOR CALCULUS (3) Let Wot - f, - SaVa « z, then Tf = V [c 2 + *% fi = — &x V • £ = #ia + c^/3 + c%y t where the subscript 1 means differentiation as to s, that is, along a line of the congruence. - S^ - cip; a! = cr'Sfei + x, or This gives the torsion in terms of the curl of a and its derivative. (4) If the curves of the congruence are normals to a set of surfaces, then a = UVu and V« = V 2 u/TVu - V(l/TVu)-Vu. Hence we have at once SaVa = = x. This condition is necessary and sufficient. (5) If also VaWoi — 0, we have a Kummer normal system of straight rays. In this case by adding the two conditions, aV\/a = 0, that is, Wot = 0. This condition is also necessary and sufficient. (6) If the curves are plane, «i = or Sa\7a = $/3V/3 + SyVy or $/?£i = — xci, which is necessary and suffi- cient. (7) If further they are normal to a set of surfaces S8VP + SyVy = = jS|8f i. The converse holds. (8) If Ci is constant, Sy£i = and conversely. If also plane, and therefore circles, #/3£i = or £i = X\a + C\x(3. This is necessary and sufficient. For a normal system of circles we have also VVa = const = C\y. (9) For twisted curves of constant curvature £i = — ciaifi. differentials 179 Notations Vortex of a vector VVu, Tait, Joly, Heaviside, Foppl, Ferraris. V X u, Gibbs, Wilson, Jaumann, Jung. curl u, Maxwell, Jahnke, Fehr, Gibbs, Wilson, Heaviside, Foppl, Ferraris. Quirl also appears. [Vm], Bucherer. rot u, Jaumann, Jung, Lorentz, Abraham, Gans, Bucherer. J rot u, Burali-Forti, Marcolongo. — ; — , Fischer. dr Vort u, Voigt. (Notations corresponding to VVu are also in use by some that use curl or rot.) Divergence of a vector — SVu, Tait, Joly. S\7u is the "convergence" of Max- well. V • u, Gibbs, Wilson, Jaumann, Jung. div u, Jahnke, Fehr, Gibbs, Wilson, Jaumann, Jung, Lorentz, Bucherer, Gans, Abraham, Heaviside, Foppl, Ferraris, Burali-Forti, Marcolongo. \7u, Lorentz, Abraham, Gans, Bucherer. — ~ , Fischer. dr Derivative dyad of a vector - SQV-u, Tait, Joly. • Vw, Gibbs, Wilson. • V ; u, Jaumann, Jung. du -p= t Burali-Forti, Marcolongo. aJr — , Fischer. dr D u - , Shaw. 180 VECTOR CALCULUS Conjugate derivative dyad of a vector — VS«(), Tait, Joly. Vm-, Gibbs, Wilson. V; u- f Jaumann, Jung. Ki(), Burali-Forti, Marcolongo. -j-, Fischer. dr c D u -,Shaw. Planar derivative dyad of a vector WVuQ, Tait, Joly. VX(mX 0), Gibbs, Wilson. V *u, Jaumann, Jung. du CK , Burali-Forti, Marcolongo. — x(D u ), Shaw. Dispersion. Concentration — V 2 , Tait, Joly. V 2 is the "concentration" of Maxwell. V 2 , Lorentz, Abraham, Gans, Bucherer. V-V, Gibbs, Wilson, Jaumann, Jung. div grad, Fehr, Burali-Forti, Marcolongo. — div grad, Jahnke. A2, for scalar operands, 1^, ,*,, A . % - . A/, for vector operands, jBurah-Forti, Marcolongo. -7-5 > Fischer. dr Dyad of the gradient. Gradient of the divergence — VSV, Tait, Joly. VV-, Gibbs, Wilson. V; V, Jaumann, Jung. grad div, Buroli-Forti, Marcolongo. DIFFERENTIALS 181 Planar dyad of the gradient. Vortex of the vortex VVVV(), Tait, Joly. V*V, Jaumann, Jung. rot 2 , Lorentz, Bucheoer, Gons, Abraham. curl 2 , Heaviside, Foppl, Ferraris. rot rot, Burali-Forti, Marcolongo. 13. Vector Potential, Solenoidal Field. If £ = VVv, then we say that a is a vector potential of £. Obviously £v£ = SV 2 <r = 0. The vector potential is not unique, since to it may be added any vector of vanishing curl. When the convergence of a vector vanishes for all values of the vector in a given region we call the vector solenoidal. If the curl vanishes then the vector is lamellar. We have an example of lamellar fields in the vector field which is determined by the gradient of any scalar function, for WVu = 0. In case the field of a unit vector is solenoidal we see from the considerations of § 12 that the first and second divergent deviations of any one of its vector lines are opposite. If then we draw a small circuit in the normal plane of the vector line at P and at the end of dp a second circuit in the normal plane at p + dp, and if we project this second circuit back upon the first normal plane, then the second will overlie the first in such a way that if from P a radius vector sweeps out this circuit then for every position in which the radius vector must be extended to reach the second circuit there is a corresponding position at right angles to it in which it must be shortened by an equal amount. It follows that the limit of the ratio of the areas of the two circuits is unity. Hence if such a vector tube is followed throughout the field it will have a constant cross- 182 VECTOR CALCULUS section. In the general case it is also clear that SVcr gives the contraction of the area of the tube. When <r is not a unit vector then we see likewise that SVcr by § 12 has a value which is the product of the con- traction in area by the TV -f- the contraction of TV multi- plied by the area of the initial circuit. Hence SVv repre- sents the volume contraction of the tube of a for length TV per unit area of cross-section. When the field is solenoidal it follows that if TV is decreasing the tubes are widening and conversely. For instance, S\/Up = — 2/Tp signifies that per unit length along p the area of a circuit which is normal to p is increasing in the ratio 2/Tp, that is, the flux of Up is increasing at the rate of 2/ Tp along p. Also £ • Vp = — 3 indicates that an infinitesimal volume taken out of the field of p is increasing in the ratio 3. Of this the increase 2 is due to the widening of the tubes, as just stated, the increase 1 is due to the rate at which the intensity of the field is increasing. If the field is a velocity field, the rate of increase of volume of an infinitesimal mass is 3 times per second. It is evident now if we multiply SVo" by a differential volume dv that we have an expression for the differential flux into the volume. If a is the velocity of a moving mass of air, say unit mass, then SV<? is the rate of compression of this moving mass, and SVcrdv is the compression per unit time of this mass, and fffSVcrdv is the increase in mass per unit time of matter at initial density or com- pression per unit time of a given finite mass which occupies initially the moving volume furnishing the boundary pf the integral. If r is the specific momentum or velocity of unit volume times the density, then SVr is the condensation rate or DIFFERENTIALS 183 rate of increase of the density at a given fixed point, and SVrdv is the increase in mass in dv per unit time. Hence SffSVrdv is the increase in mass per unit time in a given fixed space. Since 1 a — -t c where c is density at a point, SVo- = --SVct + -SVt cr c e, „ i . B log c d log c = _ S(TV . logc+ _JL_ = _iL = total relative rate of change of density due to velocity and to time, = relative rate of change of density at a moving point. SVc-dv= increase in mass of a moving dv divided by the original density. fffSVv-dv = increase in mass in a moving volume per unit of time divided by original density, = decrease in volume of an original mass. For an incompressible fluid SVcr = or a is solenoidal, and for a homogeneous fluid SVt = or t is solenoidal. In water of differing salinity #Vcr = 0, SVr =\= 0. We have a case of constant r in a column of air. If we take a tube of cross-section 1 square meter rising from the ocean to the cirrus clouds, we may suppose that one ton of air enters at the bottom, so that one ton leaves at the top, but the volume at the bottom is 1000 cubic meters and at the top 3000 cubic meters. Hence the volume outflow at the top is 2000 cubic meters. In the hydrosphere a and r 184 VECTOR CALCULUS are solenoidal, in the atmosphere r is solenoidal. We measure a in m?/sec and r in tons/ra 2 sec. At every sta- tionary boundary <r and r are tangential, and at a surface of discontinuity of mass, the normal component of the velocity must be the same on each side of the surface, as for example, in a mass of moving mercury and water. It is evident that if a vector is solenoidal, and if we know by observation or otherwise the total divergent devia- tions of a vector of length TV, then the sum of these will furnish us the negative rate of change of TV along a. Thus, if we can observe the outward deviations of r in the case of an air column, we can calculate the rate of change of TV vertically. If we can observe the outward deviations of a tube of water in the ocean we can calculate the decrease in forward velocity. EXERCISES. 1. An infinite cylinder of 20 cm. radius of insulating material of permittivity 2 [farad/cm.], is uniformly charged with l/207r electrostatic units per cubic cm. Find the value of the intensity E inside the rod, and also outside, its convergence, curl, and if there is a potential for the field, find it. 2. A conductor of radius 20 cm. carries one absolute unit of current per square centimeter of section. Find the magnetic intensity H inside and outside the wire and determine its convergence, curl, and potential. 14. Curl. We now turn our attention to another meaning of the curl of a vector. We can write the general formula for the curl W<t= -aSUaVUa- pSyVTa + y(cT(T+ SfiVTa) Let Ua = a'. These terms we will interpret, one by one. It was shown that the first term is a multiplied by the sum of the rotational deviations of <r' . But if we consider a small rectangle of sides t)dt = dip and rdu = d 2 p, then the corresponding actual deviations are Sdipd 2 a f and — Sd 2 pdia' DIFFERENTIALS 185 and the sum becomes Sdipdtff' — Sd 2 pdi<r'. But d 2 a' is the difference between the values of a' at the origin and the end of d 2 p, and to terms of first order is the difference of the average values of a' along the two sides dip and d\p + d 2 p — dip. Likewise dia is the difference between the average values of a' along the side d 2 p and its opposite. Hence if we consider Sdpa' for a path consisting of the perimeter of the rectangle, the expression above is the value of this Sdpa' for the entire path, that is, is the circulation of <j' around the rectangle. Hence the coefficient - SUaVUa is the limit of the quotient of the circulation around dip d 2 p divided by dtdu or the area of the rectangle. If we divide any finite area in the normal plane of a into elementary rectangles, the sum of the circulations of the elements will be the circulation around the boundary, and we thus have the integral theorem fSdpa = ffSdipd 2P V\7<j when Vdipd 2 p is parallel to Fy<r. The restriction, we shall see, may be removed as the theorem is always true. The component of V\7<r along a is then — Ua Lim j^Sdpcr/area of loop as the area decreases and the plane of the loop is normal to a. Consider next the term — (3SyV Ta. It is easy to reduce to this form the expression [- S*'(SyV)<r + Sy(S&'V)<r][- j8]. > id ; But this is the circulation about a small rectangle in the 13 ISC. VECTOR CALCULUS plane normal to /?. Hence the component of VVcr in the direction is — (3 Lim J'Sdpff/aresi of loop in plane normal to /?. Likewise the other term reduces to a similar form and the component of V\7<r in the direction 7 is — 7 Lim tfSdpa/sLYea of loop in plane normal to 7. It follows if a is any unit vector that the component of V\7(T along a is — a Lim JfSdpa/sLfesL of loop in plane normal to a as the loop decreases. The direction of UVS/a is then that direction in which the limit in question is a maximum, and in such case TV\7a is the value of the limit of the cir- culation divided by the area. That is, TVS/v is the maxi- mum circulation per square centimeter. Another interpretation of VV<? is found as follows: Let us suppose that we have a volume of given form and that a is a velocity such that each point of the volume has an inde- pendent velocity given by a. Then the moving volume will in general change its shape. The point which is originally at p will be found at the new point p + cr(p)dt. A point near p, say p + dp, will be found at p + dp + a(p + dp)dt, and the line originally from p to p + dp has become instead of dp, dp -f- dt[a(p + dp) — <r(p)] = dp — SdpV 'vdt. But this can be written dp' = dp- [W-^'dpa' + idpSVo- - iV(W(r)dp]dt. This means, however, that we can find three perpendicular axes in the volume in question such that the effect of the DIFFERENTIALS 187 motion is to move the points of the volume parallel to these directions and to subject them to the effect of the term dp + iV(W(r)dp dt. Now if we consider an infinitesimal rotation about the vector e its effect is given by the form (du being half of the instantaneous angle) (1 + edu)p(l — edu) = p + 2Vepdu; hence the vector joining p and p + dp will become the vector joining p + 2Vepdu and p + 2Vepdu + dp -f- 2Vedpdu, that is, dp becomes dp + 2Vedp du. We find therefore that the form above means a rotation about the vector UVV<r of amount \TV\7adt, or in other words V\/a, when a is a velocity, gives in its unit part the instanta- neous axis of rotation of any infinitesimal volume moving under this law of velocity, and its tensor is twice the angu- lar velocity. For this reason the curl of a is often called the rotation. When V\/<r = 0, a has the form a = \/u, and u is called a velocity potential. If a is not a velocity, we still call u a potential for a. EXERCISES. 1. If a mass of water is rotated about a vertical axis at the rate of two revolutions per second, find the stationary velocity. What are the convergence and the curl of the velocity? Is there a velocity potential? 2. If a viscous fluid is flowing over a horizontal plane from a central axis in such way that the velocity, which is radial, varies as the height above the plane, study the velocity. 3. Consider a part of the waterspout problem on page 50. 15. Vortices. Since VVc is a vector it has its vector lines, and if we start at any given point and trace the vector line of FVo" such line is called a vortex line. The field of FVc is called a vortex field. If a vector is lamellar the vector and the field are sometimes called irrotational. The 188 VECTOR CALCULUS equation of the vortex lines is VdpWa = - 8dp V a - V'Sdpa' - - da - V'Sdpa'. The rate of change of a then along one of its vortex lines is — V'Saa'. Since SvV^a — 0, the curl of a is always solenoidal, that is, an elementary volume taken along the vortex lines has no convergence but merely rotates. The curl of the curl is VvVVa = VV — S/SVa and thus if a is harmonic the curl of the curl is the negative gradient of the convergence, and if the vector is solenoidal, the curl of the curl is the concentration VV. EXERCISES 1. If Sa<r = = SaV '<r, and if we set <r = V-ar, and determine X so that V-X" = t, then Xa is a vector potential of the vector <r. 2. Determine the vector lines in the preceding problem for a. Also show that the derivative of X in any direction perpendicular to a is equal to the component of a perpendicular to both. What is V 2 A^? 3. If a = wy and — Sy V • w = 0, then either Xa or F/3 will be vector potentials of <r where (iy = a and all are unit vectors and SyV'X =0 = SyVY. 4. If the lines of <r are circles whose planes are perpendicular to y and centers are on p = ty, and To = f(TVyp), then any vector parallel to y whose tensor is F(TVyp), where — f = dF/dTVyp is a vector potential of a. Is a solenoidal? 5. If the lines of <r are straight lines perpendicular to y and radiating from p = ty and T<r. = f(TVyp), then what is the condition that <r be solenoidal? If Ta = /(tan -1 TVyp/Syp) a cannot be solenoidal. 6. If a =/(*Sap, S0p)-Vyp-y, then what is FV<r? Show that if/ is a function of tan -1 Sap/Spp, that SypVf is a function of the same angle, but if / is a function of TVyp, SypV •/ = and no vector of the form a = f(TVyp)Vyp-y can be a potential of yTVyp. If M = Sap/Sfip, then/0*) = - ./V0*)eW0* 2 + 1). 7. What are the lines of a = f(Sap, Sfip) Vyp and what is the curl? If / is a function of TVyp, so is the curl, and if F{TVyp) = (TVyp)- 2 fTVyp<pTVypdTVyp then F-TVyp is a vector potential of the solenoidal vector y<pT{Vyp). If / is a function of p. the curl is a function of p., and \f(ji) Vyp is a vector potential of 7/O*). 8. If <r is solenoidal and harmonic the curl of its curl is zero. If its DIFFERENTIALS 189 lines are plane and it has the same tensor at all points in a line per- pendicular to the plane, then it is perpendicular to its curl. 9. The vector <r = f- Up, where / is any scalar function of p, is not necessarily irrotational, but SaVv = 0. 10. If a vector is a function of the two scalars S\p, Sup where X, p. are any two vectors (constant), or if S\p = 0, then what is true of 11. If S<rV<r 4= 0, show that if F is determined from S\7<rVF = — SaX7 9 then F is the scalar potential of an irrotational vector r which added to <r gives a vector a', &cr'V V = 0. Is the equation for F always integrable? 12. The following are vectors whose lines form a congruence of parallel rays f(p)a, f(Sap)a, f(Vap)a, [where/ is a scalar function], which are respectively neither solenoidal nor lamellar, lamellar, solenoidal. The case of both demands that To = constant. 13. Examples of vectors of constant intensity but varying direction are o- = aUp, aVocp +«V(6 2 - a 2 V 2 ap). Determine whether these are solenoidal and lamellar. 14. If the lines of a lamellar vector of constant tensor are parallel rays, it is solenoidal. If the lines of a solenoidal vector are parallel straight lines, it is lamellar. 15. An example of vectors whose convergences and curls are equal at all points, and whose tensors are equal at all points of a surface, are a(x + 2yz) + &(y + Szx) + xyy, and 2yza + Szx/3 -f- y(xy + 2z) and the surface is x 2 + y 2 - z 2 + 6xyz = 0. Therefore vectors are not fully determined when their convergences and curls are given. What additional information is necessary to determine an analytic vector which does not vanish at oo .' Determine a vector which is everywhere solenoidal and lamellar and whose tensor is 12 for Tp m oo . 16. Show that — eV 2 <Z = lim r=0 [average value of q over a sphere of radius r, less the value at the center] divided by r 2 . — \V 2 q = average of (- SaV) 2 q in all directions a. — xVV 2 g = lim r =o [excess of average value of q throughout a small sphere over the value at the center} divided by r 2 . 17. Show by expansion that a(p + 8p) = a(p) - S8pX7 -(r(p) - VS P [- Sa8p + ±S8pV P Sa8p] - W8pVSJ p* = VVS P [~ |F5pa + iSSpVpVSpa] - ±8pSV P <r. 190 VECTOR CALCULUS i The first expansion expresses <r in the vicinity of p in terms of a gradient of a scalar and an infinitesimal rotation. The second expresses a in the form of a curl and a translation. 18. Show that for any vector <r we have £V(W'V"&r\r"<r/7V) =0, where the accents show on what the V acts, and are removed after the operation of the accented nabla. The unaccented V acts on what is left. (Picard, Traits, Vol. I, p. 136.) 19. If a, <r 2 are two functions of p, and d<n = <pi(dp),da 2 = widp), show that &<riV -SaiV — S<r 2 V -SaiV = S(<pi<r 2 — <p 2 <Ti)^7 . 16. Exact Differentials. If the expression Sadp is the differential of a function u(p), then it is necessary that Sadp = — SdpVu, for every value of dp, which gives a = — Vw. When a is the gradient of a scalar function of u(p), u is sometimes called a force-function. It is evident at once that VS7<r = 0, or £FOV)cr = for every v. This is obviously a necessary condition that Sadp be an exact differential, that is, be the differential of the same expression, u, for every dp. It is also sufficient, for if VVa = 0, it will, be shown below that a = Vu, and SVudp = — du. In general if Q(p) is a linear rational function of p, scalar or vector or quaternion, then to be exact, Q(dp) must take the form Q(dp) — — SdpV -R(p) for every dp. Hence formally we must have the identity C()= -S()V-R(p). But if we fill the ( ) with the vector form VvV , we have Q(Vi>S7) = for every v. DIFFERENTIALS 191 This may be written in the form Q'VV'l ) .- identically. EXERCISES 1. Vadp is exact only when a = a a constant vector. For VaV\7 v = for every v, that is S\(vSS7p- — VSav) = for every X, v, and for X perpendicular to v therefore SXS/Sav = 0, or Sdav = for every v perpendicular to the dp that produces da. Again if X = v, SV* + SvVSav = 0, for every v. Therefore S\/a = and Sv\7 Sav = 0, or Sdav = for every dp in the direction of v. Hence da = for every dp and a = a a constant. 2. Examine the expressions S^, V(Vap)dp, F.&. Integrating Factor If an expression becomes ezactf &?/ multiplication by a scalar function of p, let the multiplier be m. Then mQ(W) = 0, where V operates on m and Q, or QWm() + mQVV() = 0, where V operates on m only in the first term and on Q only in the second. This gives for Sadp SaVmi ) + mS( )Vo- = 0, or VaVm + mVV<r = 0. This condition is equivalent however to the condition Sa\7<7 = 0. Conversely, when this condition holds, we must have VVa- = V(tt, where r is arbitrary, hence StVv = 0, and Sa\7r = 0. But r is any variable vector conditioned only by being 192 VECTOR CALCULUS perpendicular to FV<r, hence we must have for all such VVt — 0, or a = 0. The latter is obviously out of the question and hence VVt = 0, that is t = Vw, or we may choose to write it r = Vu/u. Hence, VV<r+ VVua/u = = Vv(ua), and S(ua)dp=0 is thus proved to be exact. We may also proceed thus. Since every vector line is the intersection of two surfaces, say u = = v, then we can write the curl of a, which is a vector, in the form VV<r = hVVu\7v, and if S<tS7<t = 0, it follows that we must have a in the plane of Vw, Vfl and a = xVu -f- yVv. Sadp = — xdu — ydv. But also VVcr = VVxVu + VVyVv = hVVuVv. Hence SVuVyVv = = SVvVxVu. These are the Jacobians of u, v, x and u, v, y however, and since their vanishing is the condition of functional de- pendence, it follows that x and y are expressible as functions of u and v. Hence we have x(u, v)du + y(u, v)dv — 0. It is known, however, that this equation in two variables is always integrable by using a multiplier, say g. Therefore S(ga)dp = is exact for a properly chosen g. Further we see that ga = — Vw, or that when SaV.a = 0, a = mVw. If SVo- = for all points, then we find easily that a = Wr. For a = hVVu\/v, DIFFERENTIALS 193 so that SV<t = SvhVuVv = and h = h(u, v). Integrate h partially as to u, giving w = fhdu + f(v), then Vw = hVu + fvVv, VX/wVv = hVX/uX/v = o\ Set r = wX/v or — v X7w and we have at once a = VX/j. It is clear that if we draw two successive surfaces W\ and w 2 and two successive surfaces Vi and v 2 , since m„ Aw , „, Av T\/ w = and T\7v = Ani An 2 and the sides of the parallelogram which is the section of the tube are A<?2 = Arii esc 6, Asi = An 2 esc 6, and area = AniAn 2 esc 6, then TVx area = AwAv, and these numbers are constant for the successive surfaces, hence the four surfaces form a tube whose cross-section at every point is inversely as the intensity of a. For this reason a is said to be solenoidal or tubular. If Vx/a = for all points then we must have a = V«. For SvVo- = and a = gVv, VX/a = VX/gX/v, hence g is a function of v, and we may write a = X7u. If X7d = 0, we must have, since Sx/c = 0, a = VX/r, and since VX7(T = 0, a = X7u, whence X7 2 u — 0. Therefore, if X7(T = 0, <t is the gradient of a harmonic function and also the curl of a vector r, the curl of the curl of r vanishing. Also if VX7VX/t = 0, since we must then have Vx/t = X/v, and therefore SV^Vr = = V 2 fl, we can say that if the curl 194 VECTOR CALCULUS of the curl of a vector vanishes it must be such that its curl is the gradient of a harmonic function. Also SdpVr= —dv. Functions related in the manner of v and r are very im- portant. Since in any case SvVVcr — 0, we must have Vv<r = VVuVv or VV(<r — u\/w) = 0, whence a — uVw = Vp, so that in any case we may break up a vector a into the form a = Vp + uVw. It follows that SaV<r = SVp\7u\/w. If we choose u, w and x as independent variables, we have Vp = PxVx + p u Vu + p w Vw, whence S<tX7(t = p x SVxVuVw, and we can find p from the integral p = fSaVv/SVxVuVw-dx. In case SaVcr = 0, p = constant, and a = uVw. A theorem due to Clebsch is useful, namely that a can always be put into the form <r = Vp + VVt, where V\/Vp = 0, SVFVr = 0, that is, <r can always be considered to be due to the super- position of a solenoidal field upon a lamellar field. We merely have to choose p as a solution of V 2 p = SVcr, for we have at once Sv(<r — Vp) = 0, and therefore o- — Vp = VVt. DIFFERENTIALS 195 This may easily be seen to give us the right to set <r = Vp + (Vv) n r. EXAMPLES. SOLUTIONS OF CERTAIN DIFFERENTIAL FORMS (1). SV<t = 0, then a = VVr, and if Vv<r = 0, <r = Vp. If V<r = 0, <7 = Vh where V 2 ^ = 0. (2). If <p is a linear function dependent upon p continu- ously, and <pV = 0, <p = OVvQ- If <poV = 0, <Po = VV(6 VV0), 8, do are linear functions. For the notation see next chapter. (3). VVvQ = 0, <p = - VSaQ. If e(Fv </>()) = 0, <P = fcFVO ~ V-SerO. If (FV^())o = <W = p() - V^(). Fv^o = 0, <?o = - S()V- Vp. (4). A particular solution of certain forms is given, as follows : *SVo" = a, cr = Jap, Fv<r = eat, a = \Vap, Vp = oc, p = — Sap, yXJ = ol, (p = — Sap'Q, VV<pQ = 6, <p = - iVpdQ, €(VV<pQ) - a, ? = - &*p.(), (Fvrio = O , ip ; - - i^oO, Fw = p{), <p - - fo7p() - V&r(). EXERCISES 1. Consider the cases o- = t -\-jf(g(p)) + cfc, where/ and gr have the following values: f = g, g 2 , g 3 , <g,fg, g~\ g~ 2 , e«, log g, sin g, tan #, and g has the values y/r, (y - 'ax) /(ay + »), (bx + jf)/(a; - &y), x/y, — x/y, — y/x, etc., V (x 2 + y 2 ) — a. 2. Consider the vector lines of a = i cos (3-n-r) + j sin (3xr), r = V (x 2 + y 2 ). 3. Consider the significance of S-Ua\/Ua = 0; give examples. 4. If rf<r = Vt dp find F V <r. Likewise if da = adpd, da = aSpdp, da = —p 2 dp, da = Vradp where t is a function of p. 17. Groups. If Si, Sj, • • • , S n are any functions linear 196 VECTOR CALCULUS in V but of any degrees in p, then they form a transforma- tion group (Lie's) if and only if for any two Si, S;, where is a linear function of Si, S2, • • • S n , and a, /? arbitrary vectors. For instance, we have a group in the six formal coefficients of the two vector operators Si = - V - pSpV, S 2 = - FpV, for SaZiSpEi - S0Ei&*Ei = Sa(3Z 2 , SaE 2 S/3E 2 - £/3E 2 &*E 2 = - &x/3S 2 , &*SiS/3S 2 - S/SSt&xSi = - SapBi. The general condition may be written without a, /3 : Kt S E/ - Si'SZj - v e 0, where the accented vector is operated on by the unaccented one. Integration 18. Definition. We define the line integral of a function of p,f(p), by the expression flf(p)<p{dp) = Lim 2f( Pi )(p(dpi), % - 1, • • •, »j n = 00 where the vectors pi for the n values of i are drawn from the origin to n points chosen along the line from A to B along which the integration is to take place, <p(cr) is a function which is homogeneous in a and of first degree, rational or irrational, dpi = p t - — p z _i, and the limit must exist and be the same value for any method of successive subdivision of the line which does not leave any interval finite. Like- wise we define a definite integral over an area by the expres- sion ffi(p)<P2{dip,d 2 p) = Lim 2f(j>i)<to(dip it d 2 pi), INTEGRALS 197 where <p 2 is a homogeneous function of dipi and d 2 p{, two differentials on the surface at the point pi, and of second degree. A definite integral throughout a volume is simi- larly defined by J % J % .ff(p)<P3(dip, d 2 p, d z p) = Lim 2/(p»)¥>g(dipt, d 2 pi, d 3 pi). For instance, if we consider /(p) = a, we have for ffadp along the straight line p = fi + #7, dp = cfo-7 and Lim "Zadx-y from # = # to x = Xi is 0:7(21 — Xo), hence ^P = «(Pi - Po). The same function along the ellipse p = /3 cos + 7 sin 0, where dp = (— /? sin 6 + 7 cos 0)d0 has the limit (a/3 cos 6 -\- ay sin 0) between = O , 6 = 0i, that is, again a(pi — p ). EXAMPLES (1). j£« £dp/p = log TWpo, for any path. ( 2 )- Su ~ q~ l dqq~ l = qr 1 — g _1 , for any path. (3). The magnetic force at the origin due to an infinite straight current of direction a and intensity / amperes is H = 0.2-I-Va/p, where p is the vector perpendicular from the origin to the line. In case then we have a ribbon whose right cross-section by a plane through the origin is any curve, we have the magnetic force due to the ribbon, expressible as a definite integral, H = 0.2IfVaTdp/p. For instance, for a segment of a straight line p = a(3 -\- xy, /3, 7 unit vectors Tdp = dx, H = 0.27 / '(ay - xt3)dx/(a 2 + x 2 ) = - 0.2/0 -log (a 2 + * 2 2 )/(a 2 + *i 2 ) -f- 0.2 -I-yitsoT 1 x 2 /a — tan -1 xj/a), = 0.27/3 -log OA/OB + O.27J. L AOB. 198 VECTOR CALCULUS (4). Apply the preceding to the case of a skin current in a rectangular conductor of long enough length to be prac- tically infinite, for inside points, and for outside points. (5). Let the cross-section in (4) be a circle p — b3 — a(3 cos 6 — ay sin 0. Study the particular case when b = and the origin is the center. (6). The area of a plane curve when the origin is in the plane is \TfVpdp. If the curve is not closed this is the area of the sector made by drawing vectors to the ends of the curve. If we calculate the same integral \fVpdp for a curve not in the plane, or for an origin not in the plane of a curve we will call the result the areal axis of the path, or circuit. This term is due to Koenigs (Jour, de Math., (4) 5 (1889), 323). The projection of this vector on the normal to any plane, gives the projection of the circuit on the plane. (7). If a cone is immersed in a uniform pressure field (hydrostatic) then the resultant pressure upon its surface is "~ 2^Vpdp-P, where p is taken around the directrix curve. (8). According to the Newtonian law show that the at- traction of a straight segment from A to B on a unit point at is in the direction of the bisector of the angle AOB, and its intensity is 2/x sin ^AOB/c, where c is the perpen- dicular from to the line. (9). From the preceding results find the attraction of an infinite straight wire, thence of an infinite ribbon, and an infinite prism. (10). Find the attraction of a cylinder, thence of a solid cylinder. 19. Integration by Parts. We may integrate by parts INTEGRALS 199 just as in ordinary problems of calculus. For example, f y s V-adpSp P = iVa(B8P8 - ySfa) + \VaVPf*V pdp, which is found by integrating by parts and then adding to both sides J* y V -adpSpp. The integral is thus reduced to an areal integral. In case y and 5 are equal, we have an integral around a loop, indicated by J?. EXAMPLES (1). SfdpVcxp = HdVaS - yVay) - \Vaf*Vpdp + iSafjVpdp. (2). f y *V.VadpV(3p = ilaSPSy'Vpdp + pS-afy'Vpdp - 5 Sap 5 + y Softy]. (3). f y *S'VadpV(3p = i(Sa8S(35 - Say Spy - 8 2 SaP -y 2 Sap- S-a(3f y s Vpdp). (4). JfV-adpVPp = U*Spf y s Vpdp + pSafjVpdp - dSa(38 + y Softy + Sa5S(38 - Say Spy - 8 2 SaP + y 2 SaP - SaPffVpdp). (5). f y s SapSpdp = USadSpd - SaySPy -S-Voftf'Vpdpl (6). ffdpSap = itfSad - ySay + V-affVpdp]. (7). f y s Va P Spdp = HVadSpb - VaySPy - SoftffVpdp + PSaJfVpdp]. (8). f y s Vap-dp = i[Va6'B - Vayy + af y 8 Vpdp + SaffVpdp]. (9). fjapdp = h[a(8 2 - y 2 ) + 2af*Vpdp]. As an example of this formula take the scalar, and notice that the magnetic induction around a wire carrying a 200 VECTOR CALCULUS current of value Ta amperes, for a circular path a B - - 2p.Vap/a 2 . Therefore - fO^Sapdp/a 2 = - SfdpB = - OSfia^SafVpdp = ATafia~ 2 wr 2 . For fj, = 1, r = a, this is OAwC. This gives the induction in gausses per turn. (10). SfSdpw - i[S8cp8 - Sy<py] + SeffVpdp. (11). /^prfp = h[Vy<py - V8<p8 + <p'f*Vpdp + rmffVpdp]* (12). XVprfp = }[**.« - ^y. 7 + SeffVpdp - tp'f'Vpdp] - m.ffVpdp. For any lineolinear form SfQip, dp) = hm, «) - Q(y, y)] + ifAQiP, dp) - Q(d P , p)} = ««(*, *) - Q(r t)] + WSfVpdp. (13). State the results for preceding 12 problems for in- tegration around a loop. (14). Consider forms of second degree in p, third degree, etc. 20. Stokes* Theorem. We refer now to problem page 189, where we have the value of cro, a function of po, stated for the points in the vicinity of a given fixed point. If we write <tq for the value of a at a given origin 0, its value at a point whose vector is dp is o- = V 5p [- S<r 8p + %S8pVS(ro8p] - £F5pFVo% where V refers only to <r , and gives a value of the curl at * wii(v) = — Si(pi — Sj<pj — Sk<pk. For notation see Chap. IX. INTEGRALS 201 the origin 0. If we multiply by ddp and take the scalar, we have Sadbp = d Sp [Sa 8p - iS8pVSa dp] + iSSpd6pVV<r . Therefore if we integrate this along the curve whose vector radius is dp we have ffcSed&p = [So- 8p2 - Saodpi - §S6p 2 VS<ro8p2 + iSSpiV Saotpi] + %SW<T fVdpd8p. The last expression, however, is the value of $[FVovareal axis of the sector between dpi and 5p 2 ]. Therefore for an infinitesimal circuit we have fSvodbp = £[FVovareal axis of circuit] = SUvVVvo-dA. FWo is the curl of a at some point inside the loop. If now we combine several circuits which we obtain by subdividing any area, we have for the sum of the line integrals on the left the line integral over the boundary curve of the area in question, and for the expression on the right the sum of the different values of the scalar of the curl of a multiplied into the unit normals of the areas and the areas themselves or the area integral ffSV\/(rdipd 2 p. That is, we have for any finite loop, plane or twisted, the formula fSadp = ffSVV(TVd lP d 2 p. This is called Stokes' Theorem. It is assumed in the proof above that there are no discontinuities of a or V\/a, although certain kinds of discontinuities can be present. The diaphragm which constitutes the area bounded by the loop is obviously arbitrary, if it is not deformed over a singularity of a or V\7a. It follows that fSadp along a given path is independent of the path when the expression on the right vanishes for X4 202 VECTOR CALCULUS the possible loops, that is, is zero independently of dip, dip, or that is, V\7<r = 0. This condition is necessary and sufficient. It follows also that the surface integral of the curl of a vector over a diaphragm of any kind is equal to the circula- tion of the vector around the boundary of the diaphragm. That is, the flux of the curl is the circuitation around the boundary. We may generalize the theorem as follows, the expression on the right can be written ffSUvVVo- dA, where v is the normal of the surface of the diaphragm and dA is the area element. If now we construct a sum of any number of constant vectors a u a 2 , • • • a n each multiplied by a function of the form Saidp, Scr^dp, • • • Scr n dp, we will have a general rational linear vector function of dp, say <pdp, and arrive at the integral formula fvdp = ff<p(VUpV)dA, where the V refers now to the functions of p implied in <p. This is the vector generalized form of Stokes' theorem. If the surface is plane, Uv is a constant, say a, so that for plane paths fipdp = ff<pVVa-dA. We may arrive at some interesting theorems by assigning various values to the function <p. For instance, let <pdp = a dp, then <p(VUvV) = <t'VUvVv'=-Ui>SV<t+V'S<t'Uv+SUpV(t, whence ffS^a-dv = ffV'Sa'dv+ fVadp. If <fdp = pSdpa, INTEGRALS 203 then <pVUvV = pSUvVo- - VaUv, therefore ffV-adv = ffpSdvVa - fpSadp. If ipdp = pVdpa, tpVUvV = pV(VUvV)<r - SUva + aVv, therefore ffvdv + Sadv = - ffpV(VUvV)<r + fpVdpa, hence 2ffSadv = - ffSp(VUvV)<r + fSpdpa. EXERCISES 1. Investigate the problems of article 19, page 198, as to the applica- tion of the theorem. 2. Show that the theorem can be made to apply to a line which is not a loop by joining its ends to the origin, and after applying the theorem to the loop, subtracting the integrals along the radii from to the ends of the line, which can be expressed in terms of dx, along a line. Also consider cases in which the paths follow the characteristic lines of Vadp = 0. 3. The theorem may be stated thus: the circulation around a path is the total normal flux of the curl of the vector function a through the loop. 4. If the constant current la amperes flows in an infinite straight circuit the magnetic force H at the point p (origin on the axis) is for Tp<a H = ^IVa P , and for a<T P H = 0.2a?I/Va P , a is the radius of the wire. Then 7vH = /(a/10) inside the wire and equals zero outside. Integrate H around various paths and apply Stokes' theorem. In this case the current is a vortex field of intensity 7ra 2 7/10. 5. If we consider a series of loops each of which surrounds a given tube of vortex lines, it is clear that the circulation around such tube is everywhere the same. If the vector <r defines a velocity field which has a curl, the elementary volumes or particles are rotating, as 204 VECTOR CALCULUS we have seen before, the instantaneous axis of rotation being the unit of the curl, and the vector lines of the curl may be compared to wires on which rotating beads are strung. It is known that in a perfect fluid whose density is either constant or a function of the pressure only, and subject to forces having a monodromic potential, the circulation in any circuit through particles moving with the fluid is constant. [Lamb, Hydrodynamics, p. 194.] Hence the vortex tubes moving with the fluid (enclosing in a given section the same particles), however small in cross- section, give the same integral of the curl. It follows by passing to an elementary tube that the vortex lines, that is, the lines of curl, move with the fluid, just as if the beads above were to remain always on the same wire, however turbulent the motion. In case the vortex lines return into themselves forming a vortex ring, this leads to the theorem in hydrodynamics that a vortex ring in a perfect fluid is indestructible. It is proved, too, that the same particles always stay in a vortex tube. 6. Show that for a- = a(3S 2 ot P - 2Sp P ) + £(4# 3 /Sp - 2Sa P ), where Sa& = 0, the integral from the origin to 2a -J- 2/3 is independent of the path. Calculate it for a straight line and for a parabola. 7. The magnetic intensity H, at the point 0, from which the vector p is drawn to a filament of wire carrying an infinite straight current in the direction a, of intensity I amperes, is given by H = 0.27/Fap. Suppose that we have a conductor of any cross-section considered as made up of filaments, find the total magnetic force at due to all the filaments. Notice that H = 0.2/ Fa V log TVap, and that a is the unit normal of the plane cross-section of the conductor. Hence ffHdA = ff0.2IVaV log TVapdA = f0.2I log TVapdp around the boundary of the cross-section. This can easily be reduced to the ordinary form 0.21 j? log rdp. This expression is called a log- arithmic potential. If I were a function of the position of the filament in the cross-section, the form of the line-integral would change. For a circular section we have the results used in problem 4. Con- sider also a rectangular bar, for inside points and also for outside points. 8. If or and r are two vector functions of p, we have the theorem SVUuVVo-t = St(VUvV)* - S<t(VUpV)t, whence ffSr(VUpV)o- = ffS<r{VUvV)r + fSdpar, INTEGRALS 205 for a closed circuit. Show applications when a or t or both are sole- noidal. 9. Show that ffS-dvotS\7<x = fSdpaa + ffSdv(SaV)<r, ffS-Vuadv = JTuSadp - f fuSV adp, ffS-X7uS7vdv= fuSsjvdp = - fvS\7ud P , f 1 hiSVvd P = [uv] p p l - f^vSS/udp. 10. Prove Koenig's theorems and generalize. (1) Any area bounded by a loop generates by translation a volume = — Saw, where co is the areal axis; (2) The area for a rotation given by (a + Vap)at is — J] Saco + f t Scf VpVpdp. 21. Green's Theorem. The following theorem becomes fundamental in the treatment of surface integrals. Refer- ring to the second form in example 17, page 189, for the expression of a vector in the vicinity of a point, which is 0" = FV Sp [- iV8pa + iS8pVVdp<ro] - l&pSV<To we see that if we multiply by a vector element of surface, Vdi8pd 2 dp, and take the scalar Scdrfpdidp = SUvVs p []dA - iSV(r Sd l dpd 2 dp8p. If now we integrate over any closed surface the first term on the right gives zero, since the bounding curve has be- come a mere point, and thus, indicating integration over a closed surface by two J', j> $&<jd\hpd<ihp = — \S\7(TQjf jfSdibpd'ibpbp. But the last part of the right hand member is the volume of an elementary triangular pyramid whose base is given by didpd 2 8p. Hence, the integral is the elementary volume of the closed surface, and may be written dv, so that we have for an elementary closed surface j> \fSad18pd2dp = SVvodv. 206 VECTOR CALCULUS If now we can dissect any volume into elements in which the function has no singularities and sum the entire figure, then pass to the limit as usual, we have the important theorem ffS<rd lP d 2 p = fffSVv dv. This is called Greens theorem, or sometimes Green's theorem in the first form. It is usually called Gauss' theorem by German writers, although Gauss' theorem proper was only a particular case and Green's publication antedates Gauss' by several years. The theorem may be stated thus: the convergence of a vector throughout a given volume is the flux through the bounding surface. It is evident that we can generalize this theorem as we did Stokes' and thus arrive at the generalized Green's theorem $ fQvdA = f f f$\/ dv. v is the outward unit normal. The applications are so numerous and so important that they will occupy a considerable space. • The elementary areas and volumes used in proving Stokes' and Green's theorems are often used as integral definitions of convergence or its negative, the divergence, and of curl, rotation, or vortex. For such methods of approach see Joly, Burali-Forti and Marcolongo, and various German texts. A very obvious corollary is that if SVc = then $ \fSad1pd2p = 0. It follows that the flux of any curl through any closed sur- face is zero. Hence, if the particles of a vortex enter a closed boundary, they must leave it. Therefore, vortex tubes must be either closed or terminate on the boundary wall of the medium in which the vortex is, or else wind about infinitely. We may also state that if SVa = the differential expression Sadipd 2 p is exact in the sense that INTEGRALS 207 J % J % S(rdipd 2 p is invariant for different diaphragms bounded by a closed curve, noting the usual restrictions due to singularities. We proceed to develop some theorems that follow from Green's theorem. Let $Uv be — pSUvcr, then 3>V = — pSv<r + o- and we have fffadv = fffpSVvdv - ffpSUvadA. Let $Uv = — pVUva, then <i>V = — pVVv + 2a and SSfvdv = ifffpVVvdv - \ffpVUvodA. Let $Uv = pSpUva, then <J>V = pSpVv + Fpo-, whence fffVpa dv = - fffp&Va dv + ffpSpUvadA. Let $17V = - pVpVUixr, then $V = - pFpFVo" + 3PV, hence SSSVpadv = ifffpVpWadv- \ffpVpVUvadA. Let $E7V = SprUiHT, then 3>V = SprV<r + Spa\/r + Sot, thence fffSar dv = - fffiSprV* + &rVr)<fo + f f&prTJva dA. In the first of these if a- has no convergence we have the theorem that the integral of cr, a solenoidal vector, through- out a volume is equal to the integral over the surface of p multiplied by the normal component of a. In the second we have the theorem that if the curl of a vanishes through- out a volume, so that a- is lamellar in the volume, then the integral of a throughout the volume is half the integral over the surface of p times the tangential component of a taken at right angles to a-. In the third, if the curl of cr 208 VECTOR CALCULUS vanishes then the integral of the moment of a with regard to the origin is the integral over the surface of Tp 2 times the component along p of the negative of the tangential com- ponent of a taken perpendicular to <r, and by the fourth this also equals the surface integral of the component perpendicular to p of the negative tangential component of <r taken perpendicular to a. According to the fifth formula, if a solenoidal vector is multiplied by another and the scalar of the product is integrated throughout a volume, then the integral is the integral of — SpaVr throughout the volume -f- the integral of ScrprUv over the surface. If in the first, second, third, and fourth we set c<t for a, and in the fifth ca for a and — \<t for r, we have from the first and second the expression for X, the momentum of a moving mass of continuous medium, of density c, and from the third and fourth the moment of momentum, /x, and from the fifth the kinetic energy. If the medium is in- compressible, and we set 2k = V\/v, since SVca = 0, then X = fffcadv = - ffcpSUvadA + fffpSaVcdv + SSfcpSV* dv = fffpcKdv+lfffpWcadv - \££cpVVvadA. ju = fffcVpadv = ffcpSpUvadA - SSfcpSpVa - fffpSpVca = UffcpVpK + \fffpVpWcadv - \ffcpVpVVvadA. T = - hSfSSa 2 cdv = - hffSpvUvac dA + SffhcSpaVo- dv + hfffSpvVca dv. In case c is uniform these become still simpler. If we set a = S/u and r = \/w in the above formula we INTEGRALS 209 arrive at others for the gradients of scalar functions. The curls will vanish. If further we suppose that u, or w, or both, are harmonic so that the convergences also vanish we have a number of useful theorems. Othei forms of Green's theorem are found by the follow- ing methods. Set $Uv = uS\7wUv, then $V = u\/ 2 w + SVuVw and we have the second form of Green's theorem at once SfS&VuVw dv = ffuS\/wUv dA — fffu\7 2 wdv, and from symmetry yWSvWw dv = ffwSVuUv dA — fffw\/ 2 u dv. Subtracting we have J % J *J % (u\7 2 w — w\7 2 u) dv = ~ f£(,STJv[u\7w - wVu])dA. 22. Applications. In the first of these let u = 1, then fffV 2 ™ dv — — J'.fSUj'VwdA. If then w is a har- monic function, the surface integral will vanish, and if V 2 w = 47Tju, which is Poisson's equation for potentials of forces varying as the inverse square of the distance, inside the masses, ju being the density of the distribution, then ffSUvS7w dA = ±ttM, where M is the total mass of the volume distribution. This is Gauss' theorem, a particular case of Green's. In words, the surface integral of the normal component of the force is — 47r times the enclosed mass. The total mass is l/4x times the volume integral of the concentration. In the first formula let u = 1/Tp and exclude the origin 210 VECTOR CALCULUS (a point of discontinuity) by a small sphere, then we have fffSV(l/Tp)Vwdv = ffdA SUrVw/T P - fffdv V 2 w/Tp for the space between the sphere and the bounding surface of the distribution w, and over the two surfaces, the normals pointing out of the enclosed space. But for a sphere we have dA = Tp 2 dw where co is the solid angle at the center, and dv = Tp 2 dwdTp. Thus we have fffV 2 w/Tp dv = ffSdA UuVw/Tp -fffSv(l/T P )Vwdv = ffSdA UvVw/Tp -fffSv(wV[l/T P ])dv since V 2 l/7p = 0, = ffSdA UvVwjTp -ffSdA UvwV(l/Tp) = ffSdA UvVwjTp+ffSdA VvVp\T 2 pw. Now of the integrals on the right let us consider first the surface of the sphere, of small radius Tp. The first integral is then - ffSUpX/wlTp- T 2 pdco = - ffSUpVw- Tpda, and if we suppose that the normal component of Vw, that is, the component of Vw along p, is everywhere finite, then this integral will vanish with Tp. The second integral for the sphere is — J?rf'SUpUpwT 2 pd(x)lT 2 p = — tfj'wdu, and the value of w at the origin is Wo, then this integral is 47TWo since the total solid angle around a point is 47r. Hence we have fffdv V 2 w/Tp = ffSUv{\/wlTp + wUp/T 2 p)dA + 4twq and 4x^o= fffdvV 2 w/Tp - ffSUp(Vw/Tp -f wUp/T 2 p) dA, INTEGRALS 211 where the volume integral is over all the space at which w exists, the origin excluded, and the surface integral is over the bounding surface or surfaces. In words, if we know the value of the concentration of w at every point of space, and the value of the gradient of w and of w at every point of the bounding surfaces at which there is discontinuity, then we can find w itself at every point of space, provided w is finite with its gradient. If X7 2 w is of order in p not lower than — 1 we do not need to exclude the origin, for the integral is ///V 2 ^ TpdcpdTp, and this will vanish with Tp when V 2 w is not lower in degree than — 1. EXERCISES 1. We shall examine in detail the problem of w — const, over a given surface, zero over the infinite sphere, V 2 w = everywhere, \/w = on the inside of the sphere, but not zero on the outside. Then for the inside of the sphere the equation reduces to 4:irw = - £fwSUvUplT*pdA = 4ttu;, hence w is constant throughout the sphere and equal to the surface value. On the outside of the sphere, we have to consider the bounding sur- faces to be the sphere and the sphere of infinite radius, so that we have 4^0 = _ ffSdA UvVw/Tp- wffSdA UuUpfTp 2 , where the first integral is taken over both surfaces and the second integral is over the given surface only, since w = at °° . The second integral vanishes, however, since it is equal to w times the solid angle of the closed surface at a point exterior to it. If we suppose then that \/w is at «3 we have a single integral to evaluate 4:irw = — j> j> 'SdAU r i>\? 'w/T 'p over the surface. A simple case is — SUv\/w = const. = C. Then 4ttWo = CffdAITp. The integration of this and of the forms arising from a different assump- tion as to the normal component of V^ can be effected by the use of fundamental functions proper to the problem and determined by the boundary conditions, such as Fourier's series, spherical harmonics, and the like. One very simple case is that of the sphere. If we take 212 VECTOR CALCULUS the origin at the center of the sphere we have to find the integral ,f,fdA/T( P - P o) where po is the vector to the point. Now the solid angle subtended by po is given by the integral — r~ l ffdASpU{p — po)/T*(p — p ) and equals 4t or 0, according as the point is inside or outside of the sphere. This integral, however, breaks up easily into two over the surface, the integrands being r-^T-Kp - po) - SpoU( P - P0 )/T*(p - po), but the last term gives or — 47rr 2 /7 T p , as the point is inside or outside of the sphere. Hence the other term gives ffdAlT{p - po) - 47rr or 4Trr 2 /Tp as the point is inside or outside. We find then in this case that w m Cr 2 /Tpo. If in place of the law above for — SUvS7w, it is equal to C/T 2 (p — p ) we find that ffdAIT\ P - po) = 47rr/(r' + p 2 ) or 47^/(7^0 - r^po). Inside _ r r ,A S(p ~ pp)(p + po) - ffdA TKp - po) ' dA = 27rr 2 sin Odd =- d[a 2 + r 2 - x 2 ] = —xdx, a a „ po(p — po) = ax cos 4/ _ a 2 + x 2 — r 2 T*(p - po) " x 3 2x* ffdAS^f^=^f r+a ' a+r ( a ^ + l)dx = T 2 (p — po) aJr-a,a-r \ X 2 J or 47TT 2 a The differentiation of these integrals by using Vp as operator under the sign leads to some vector integrals over the surface of the sphere. 2. Show that we have ££UvdAIT(p - po) = |ttpo or |7rr 3 /^ 3 Po-po for inside or outside points of a sphere. INTEGRALS 213 3. Find ffdAUu/T 3 ( P - Po ) for the sphere. 4. Prove f fdAT^{p-fi)T-\p-oc) =4 7 rr/[(r 2 -a 2 )7 7 ( / S-«)] or = ^r 2 J[a(r 2 -a 2 )T(r 2 a- 1 +0)]. 5. Consider the case in which the value of w is zero on a surface not at infinity but surrounding the first given surface. We have an example in two concentric spheres which form a condenser. On the inner sphere let w be const. = Wi, on the outer let w = 0, on the inner let — SUpVw = 0, inside, = E h outside, on the outer let — SUv\/w = E 2 on the inside, = Oon the outside. 6. If w is considered with regard to one of its level surfaces, it is constant on the surface, and the integral — £ f SdAU vU p\T 2 pio becomes for any inside point 4:irw, hence we have 4irw - A.™ = fffdv\7 2 wlT P - £ £SdAUuVw/T P . If then w is harmonic inside the level surface, it is constant at all points and 47r(w - to) m - £fSdAUv\7wlTp. But since w is constant as we approach the surface, V^o =0, and V(w — Wo) = 0, so that X7w = 0. Hence w = w. If w vanishes at oo and is everywhere harmonic it equals zero. 7. If two functions Wi, w 2 are harmonic without a given surface, vanish at » , and have on the surface values which are constantly in the ratio X : 1, X a constant, then W\ = \W2. 8. If the surface Si is a level for both the functions u and w, as also the surface S 2 inside Si, and if between Si and $2, u and w are harmonic, then (U — Ui)(w 2 — Wi) = (W — Wi)(ll2 — Ui). For if w = <p(u), then V 2 w = = <p"(u)T 2 \7u, hence <p(u) — au + b, etc. [A scalar point function w is expressible as a function of another scalar function u if and only if V\/w\7u = 0.] 9. Outside a closed surface S, Wi and w 2 are harmonic and have the same levels. Si vanishes at • while w 2 has at 00 everywhere the con- stant value C. Then w 2 = Bwi + C. For Vw 2 = tVw h V 2 w 2 = V^V^i = 0, thus V* = 0, or V^i = 0, and t = B or wi = const. 10. There cannot be two different functions W\, w 2 both of which within a given closed surface are harmonic, are continuous with their gradients, are either equal at every point of S or else SUvX/Wi =SUv\/w 2 at every point of S while at one point they are equal. Let u = Wi — w 2 , then V 2 w = 0, SJu = on S or else SUv\/u = 0, and at one point Vw = 0. 214 VECTOR CALCULUS 11. Given a set of mutually exclusive surfaces, then there cannot be two unequal functions w\, Wi, which outside all these surfaces are harmonic, continuous with their gradients, vanish at <» as Tp~ l , their gradients vanishing as Tp -2 , and at every point of the surfaces either equal or SUvVwi = SUvVwt- 23. Solution of Laplace's Equation. The last problems in the preceding application show that if we wish to invert V 2 w = 0, all we need are the boundary conditions, in order to have a unique solution. In case V 2 u is a function of P>f(p)> we can proceed by the method of integral equations to arrive at the integral. However the integral is express- ible in the form of a definite integral, as well as a series, w = l/4:w[fSSdvV 2 w/Tp - ffSUviVw/Tp + wUp/T 2 p)dAl The first of these integrals is called the potential and written Pot. Thus for any function of p whatever we have Vot q, = fffqdvlT(p- p Q ) where p describes the volume and p is the point for which Pot qo is desired. Let Vo be used to indicate operation as to po, then we have Vo Pot g = VoffSqdv/T(p - p ) = fff[dvU(p - p )/r 2 (p - Po )]q - -SSfV[qlT(p- p )]dv + SffdWq/T(p- po) = Pot Vg - ffdAUvqlT(p - Po ). If we operate by Vo again, we have Vo 2 Pot q = Pot V 2 ? - ffdA[Uv\7qlT(p - po) + V'Uvq/T'(p - po)]. But the expression on the right is 4x^0, hence we have the INTEGRALS 215 important theorem Vo 2 Pot q = 4:irq . That is, the concentration of a potential is 4x times the function of which we have the potential. In the case of a material distribution of attracting matter, this is Poisson's equation, stating that the concentration of the potential of the density is 4r times the density; that is, given a distribution of attracting masses, they have a potential at any given point, and the concentration of this potential at that point is the density at the point -5- 4-7T. The gradient of Pot q was called by Gibbs the Newtonian of g , when the function q is a scalar, and if q is a vector, then the curl of its potential is called the Laplacian, and the convergence of its potential is called the Maxwellian of q . Thus New q = Vo Pot P, Lap <r = V\/o Pot <r , Max (To = £ Vo Pot co. We have the general inversion formula 47rVo~ 2 Vo 2 ? = 47rgo - SSfV 2 q/T(p - Po )dv - ffdA[UvTqlT{p - p ) + U(p - p )qUr/T*(p - p )J. This gives us the inverse of the concentration as a potential, plus certain functions arising from the boundary conditions. We may also define an integral, sometimes useful, called the Helmholtzian, Him. Q m fffQT{p - Po )dv. Certain double triple integrals have been defined: Pot 0, v) = ffffffu(p 1 )v(p2)dv 1 dv 2 /T(p 1 - p 2 ), 216 VECTOR CALCULUS Pot (to) = fffffS - Sh dvidvJTfa - p 2 ), Lap (to) = ffffff + 5to(Pi - p 2 )^i^ 2 /P( Pl -p 2 ), New («, f) = SSSSSS-S{i(pi-p2)v l dvidv 2 IT'(pi-p t ), Max(£,*) = - ffffffv l SUpi-P2)dv l dv 2 ir( Pl - P2 ). EXERCISES 1. Iff = — VP is a field of force or velocity or other vector arising from a scalar function P as its gradient, then Po = - SSSSV£dv/[4irT( P - po)] + ffdA[SUvll&*T{p - po)) + PC/ y V^(p-po)/47r]. If P is harmonic the first term vanishes, if £ = the first two vanish. 2. If £ = V<r, that is, it is a curl of a solenoidal vector, °o = fffVV* dv/[4irT(p - po)] - f<fdA[VUv<rl[±TcT{p - po)] + U(p - p 9 )<rlU,[4*T*(j> - po)]. 3. We may, therefore, break up (in an infinity of ways) any vector into two parts, one solenoidal and the other lamellar. Thus, let a = 7T + t where £v r ■ 0, and Wir = 0, then Sv <r = SVx and since VVt = 0, this may be written Vt = &Vo" whence x. VVc = FVt = Vr whence t. We have, therefore, from these two 47T<r = VfffSdvV<r/T(p - Po ) - V £ £SdAUvalT{p - Po ) + V jffPSdA UvV (UT(p - po) + V V SSfVV*dv/T(p - Po ) -WffVU*adA/T(p-po) + VVffDSU*S7(p+( P -po)dA t where P is such that V 2 P = *S'V<r and D such that \7 2 D = Vs/a. 3. Another application is found in the second form of Green's theorem. According to the formula SffiyW 1 ™ - wV 2 u) dv = - tf£(SUv[u\7w - w\/u])dA it is evident that if G is a function such that V 2 G = 0, and if, further, G has been chosen so as to satisfy the boundary condition SUvS7G = 0, then the formula becomes SffGs^wdv = - ££SUv\7wGdA. If then V 2 w is a given function we have the integral equation JfGSUvVwdA - - fffGj{ P )dv. Similar considerations enable us to handle other problems. 4. If u and w both satisfy V 2 / = 0, then we have Green's Reciprocal Theorem: ffuSUvSfw dA = ffwSUvVudA, Thus let therefore INTEGRALS 217 ff ^p dA = ffuSUvV -L dA. 5. Let A relate to a as V to p ; then A Pot Q = ff/QdvU(p - a)/T*( P - a) = fffV(Q/T(p - cc))dv + fffdWQ/T(p - a) = Pot VQ - ffdAUuQ/T( P - a). If Q — on the surface, the surface integral = 0. New P = Pot V - ffVvP dAjT{p - a) = A Pot P when Pot exists. Lap a = V Pot Vo- - ££VTJvadA\T{p - a) = VA Pot a when Pot exists. Max o- = S Pot V<r - £ fSUvadAITip -a) = SA Pot o- when Pot exists. A 2 PotQ = Pot V 2 Q - ££U v \7QdAIT(p - a) + //diVi[^/7 7 i( P -«)]. If Q = on the surface, that is, if Q has no surface of discontinuity, A 2 Pot Q - Pot V 2 Q, A New P = A 2 Pot P, A Lap o- = A7A Pot a, A Max a = A/SA Pot <r. 6. If j8 is a function of the time t, then d--^[yy/i(rvFv^+M^J t+br r r + VV £<fj Vdu p t+br - ff Vdv WPt+br where the subscript means t + br is put for t after the operations on have occurred. 15 CHAPTER IX THE LINEAR VECTOR FUNCTION 1. Definition. If there is a vector a which is an integral rational function <p of the vector p, a = <p' P , and if in this function we substitute for p a scalar multiple tp of p, then we call the vector function a linear vector func- tion if a becomes ta under this substitution. It is also called a dyadic. The function <p transforms the vector p, which may be in any direction, into the vector <r, which may not in every case be able to take all directions. If p = a, then we have (pp = <pa, and <p as an operator has a value at every point in space. We may, in fact, look upon <p as a space trans- formation that deforms space by a shift in its points leaving invariant the origin and the surface at infinity. In the case of a straight line Vap = /?, or p = xa + cT l (5, we see that the operation of <p on all its vectors gives a = x<pa + (pVa~ 1 ^ f and this is a straight line whose equation is Vipaa = V<pa(pVa~ 1 ^, which will later be shown to reduce to a function of (3 only, <p(3. Hence <p converts straight lines into straight lines. The lines a for which Vacpa = 0, remain parallel 218 THE LINEAR VECTOR FUNCTION 219 to their original direction, others change direction. Again if we consider the plane S-afip = or p = xa + y(3, , a = x<pa + y<pf}, so that S(r<pct<pp = 0. Hence planes through the origin, and likewise all planes, are converted into planes. These will be parallel to their original direction if Va(3 = uV<pa<p(3, or VVa$V(pa<p& = = Scx<pa<p(3= S(3<pa<p(3= Sa(3<pa = So@<pP. Now Va(3 is normal to the plane, and /3 is any vector in the plane, and <p(3 by the equation is normal to Vafi, hence <p(3 = va + w(3 for all vectors in the plane. Since <p0 = 0, the function leaves the origin invariant. Consequently the lines and planes through the origin that remain parallel to themselves are invariant as lines and planes. These lines we will call the invariant lines of <p, and the planes the invariant planes of tp. 2. Invariant Lines. In order to ascertain what lines are invariant we solve the equation Va<pa = 0, or (pa = ga, that is (tp - g)a = 0. First we write a in the form aS\fiv = \SfJiva + ixSvka + vSXfxa, where X, ju, v are any three noncoplanar vectors. Then we have at once (<p — g)\Sixvicx + (<p — g)nSv\a + (<p — g)pS\fxa = 0. 220 VECTOR CALCULUS But this means that we must have for any three non- coplanar vectors X, /i, v S(<p - g)\(<p - g)fi(<p - g)v = = tfSXiiv — g 2 (S\ii<pv + S\<ptxi> + S<p\nv) + g(S\(pfJL(pV + S\jJl<pV + S(f\(pfJLP) — S<p\<piJ.<pi>, an equation to determine g, which we shall write g z - mig 2 + m 2 g - m 3 = 0, called the /a<6n< equation of #>, where we have set Wl = (S\jA<pV + S\<pflP + S<p\fAl>)/S\fJLl>, rri2 = (S\(pii(pp -+- S<pkyupv + S(p\<piJLp)lS\fjLv, These expressions are called the nonrotational scalar in- variants of <p. That they are invariant is easily seen by substituting X' + v/jl for X. The resulting form is precisely the same for X r , ju, p, and from the symmetry involved this means that for X, /x, v we can substitute any other three noncoplanar vectors, and arrive at the same values for mi, m2, m 3 . It is obvious that m 3 is the ratio in which the volume of the parallelepiped X, jjl, v is altered. If m 3 = one or more of the roots of the cubic are zero. The number of zero roots is called the vacuity of (p. If is obvious that the latent cubic has either one or three real roots. 3. General Equation. We prove now a fundamental equation due to Hamilton. Starting with <p we iterate the function on any vector, as p, writing the successive results thus p, <pp, <p<pp = <p 2 p, <p<p<pp = <p<p 2 p = <p 3 p, "•. We have then for any three vectors X, ,u, v that are not coplanar THE LINEAR VECTOR FUNCTION 221 S\pi>(<p 3 p — mi<p 2 p) = (p 2 (<pp — m\p)S\pv = <p 2 [<p\Spvp -\- • - - — pSpv(p\ — • • •] = - <p 2 [V-VppV<p\p+ •••] = <p\V'V<p\pVp.v+ •••] = <p[<p\Sv<ppp + <pp<S\<pvp + <pvSp<p\p — <p\SpL(pvp — (fpSvcpXp — <pi>S\(pfxp]. Adding to this result S\pu> -m%ipp, we have S\pv((p 3 p — mnp 2 p + m<npp) = <p[\S<pfi<pvp + pSipVipkp + vS<p\(pp,p] = pS(f\cppapv. Subtracting SXfMV-rritp from both sides and dropping the nonvanishing factor S\p,i>, we have the Hamilton cubic for <p <p s p — mi<p 2 p + m*<pp — mzp = 0. This cubic holds for all vectors p, and hence, may be written symbolically <p 3 — mnp 2 + m 2 (p — ra 3 = identically. This is also called the general equation for <p. It is the same equation so far as form goes as the latent equation. Hence we may write it in the form (<p - gi)(<p — g*)(<p — gz) = 0. In other words, the successive application of these three operators to any vector will identically annul it. We scarcely need to mention that the three operators written here are commutative and associative, since this follows at once from the definition of linear vector operator, and of its powers. It is to be noted, too, that <p may satisfy an equation of lower degree. This, in case there is one, will be called the characteristic equation of <p. Since <p must satisfy its general 222 VECTOR CALCULUS equation, the process of highest common divisor applied to the two will give us an equation which <p satisfies also, and as this cannot by hypothesis be lower than the char- acteristic equation in degree and must divide it, it is the characteristic equation. Hence the factors of the char- acteristic equation are included among those of the general equation. We proceed now to prove that the general equation can have no factors different from the factors of the characteristic equation. (1) Let the characteristic equation be (<p - g)p = for every vector; then assuming any X, /x, v, we find easily for the latent equation x*-Sgx 2 +3g 2 x-g*= 0, so that the general equation is (cp - gf = 0. In this case if = [g\SM) + gpSrkQ + gpSlnOV&V, where X, /z, v are given for a given <p. (2) Let the characteristic equation be (<P - 9i)(<P - 92) P = 0, then by hypothesis, there is at least one vector a for which we have (<p - gi)a + 0, and at least one fi for which (<p - gt)0 4= 0. Let us take then O - gi)a = X, (<p— g 2 )(S = M- THE LINEAR VECTOR FUNCTION 223 Then (<p - g 2 )\ = 0, (<p- gi)fx = 0. Hence, we cannot have X and ju parallel, else gi = g 2 , which we assume is not the case, since from (<p- g 2 )U\ = 0, (<p- g 1 )U f x= 0, we have g 2 U\ = giUn, and g 2 = g u if X is parallel to /z, that is if U\ would = Up, There is still a third direction independent of X and /z, say v. Let cpv = av + bjjL + cX. Then we have (<p - ft)* = (a - gi)j>+ bfx + cX. Since (<p- fc)(* - 9i)v = 0, (a - gx)(<p - g 2 )v — b(g 2 - fi)p = = (a— g Y ){a — g 2 )v + 6 (a - g 2 )fx + c(a — g{)\. We must have, therefore, either a = gi and 6=0, or a = g 2 and c = 0. As the numbering of the roots is immaterial, let us take a = gi t b = 0, then <pv = giv + cX, <pX = # 2 X, ^>m = 9iV> We notice that if c # 0, we can choose v' = v — (cjg 2 )\, whence ipv' = giv' and we could therefore take c = 0. Hence g 3 - g\2gi + g 2 ) + ^(2fir^ 2 + g Y 2 ) - g?g 2 = 0, <p = [guiS\vQ + givSXpQ + # 2 X£mK)]ASX/xj>, 221 VECTOR CALCULUS and the general equation is (<P - 9i) 2 (<P ~ 92) = 0. (3) Let the characteristic equation be (<p - g)*p = 0. Then there is one direction X for which we have <p\ = g\, and there may be other directions for which the same is true. There is at least one direction \i such that (cp - g)fi = X. We have, therefore, <PV = g» + X <?X = g\. Let now v be a third independent direction, then we have (pv = av + bjj. + cK, (<p - g)v = (a — g) v + 6/x + c\, (<p - gfv = = (a - gfv + b(a - g)p + [b + c(a - g)]K. Therefore, we have a = g, 6=0, <pp = gv + cX and <£>(*> — c/x) = g{y — c/jl) = gv' , and the general equation i* - g) 3 = 0, <p = g + XSj/XO/SX/x*'. We are now in a position to say that the general equation has exactly the same factors as the characteristic equation. Further we can state as a theorem the following: (a) // the characteristic equation is of first degree, O - g Y )p = 0, then every vector is converted into g\ times that vector, by the operation of (p. THE LINEAR VECTOR FUNCTION 225 (6) // the characteristic equation is of the form O - 9i)(<P ~ 92) = 0, then there is one direction X such that <pk = 92K, while for every vector in a given plane of the form x\i.-\- yv we have (<p- #i)Om + yv) = 0. Hence <p multiplies by gi every vector in the plane of /a, v, and by g 2 all vectors in the direction X. (c) If the characteristic equation is W - g,f = 0, there is a direction such that <p\ = gi\ and a given plane such that for every vector in it x\x-\- yv we have (<P — 9i)(w + yv) = ^X. If (<P — gi)v = v\ (<P — gi)v = w\ we may set • w giving (p/i = gip. Therefore <p extends all vectors in the ratio gi, and shears all components parallel to v in the direction X. 4. Nondegenerate Equations. We have left to consider the three cases (<p — 9i)(<p — 92) (<p - gz) = 0, O - gi) 2 (<P - 02) = 0, (v - g,f = 0. In the last case we see easily that there is a set of unit vectors X, ju, v such that 226 VECTOR CALCULUS <p\ — g{K + mo, <PH = giii + vb, <pv = giv. Hence we see that <p(x\ + yi* + zv) = gi{x\ + y\x + zv) + a*M + 6y*> = gi(x\ + 2/M + •*) + a(a*M + 0*0 + (6 - a)yy, <p(x» + yv) = gi(xn + ?/*>) + fo^, <p = gi + [apSuvQ + bvSv\Q]/S\nv. Therefore <p extends all vectors in the ratio g\, shears all vectors X in the direction of m> and all vectors /x in the direction v. In the first case we see that there is at least one vector p such that {<P - 9\){<P - 9s) P = A, where <p\ = g{K. Likewise there are vectors that lead to /x and v where <PH = g 2 n, <pp = gzv. These are independent, and there- fore if we consider any vector p = x\ + yn + zp, we have <pp = xg{K + 2/02M + zg 3 v, <p = [g^SfivQ + fwJSrhO + gsvS\nO]lS\fiP. Evidently we can find X, /x, v by operating on all vectors necessary in order to arrive at nonvanishing results by (<P — 9z)(<P — 9s), (<P — 9i)(<P — 9*)> (<P — 9\)(<P — 9t) respectively. In the second case, we see in a similar manner that there THE LINEAR VECTOR FUNCTION 227 are three vectors such that £>X = g{K + \x, <pfi = giii, <pv = g 2 p, <P = IgiQiSpvQ + nSvkQ + fiwStoQ + jtSMW&Vv. 5. Summary. We may now summarize these results in the following theorem, which is of highest importance. Every linear vector function satisfies a general cubic, and may also satisfy an equation of lower degree called the char- acteristic equation. If the equation of lowest degree is the cubic, then it may have three distinct latent roots, in which case there corresponds to each root a distinct invariant line through the origin, any vector in each of the three directions being extended in a given ratio equal to the corresponding root; or it may have two equal roots, in which case there corresponds to the unequal root an invariant line, and to the multiple root an invariant plane containing an invariant line, every vector in the plane being multiplied by the root and then affected by a shear of its points parallel to the invariant line in the plane; or there may be three equal roots, in which case there is an invariant line, a plane through this line, every line of the plane through the origin being multiplied by the root and its points sheared parallel to the invariant line, and finally every line in space not in this plane is multiplied by the root and its points sheared parallel to the invariant plane. In case the function satisfies a reduced equation which is a quadratic, this quadratic may have unequal roots, in which case there is an invariant line corresponding to one root and an invariant plane corresponding to the other, any line in the plane through the origin being multiplied by the corresponding root; or there may be two equal roots, in which case there is an invariant plane such that every line in the plane is multiplied by the root and every vector not in the plane is multiplied by the root and its points displaced parallel to an invariant line. In case 22$ VECTOR CALCULUS the reduced equation is of the first degree, every line is an invariant line, all vectors being extended in a fixed ratio. Where there are displacements, they are proportional to the distance from the origin, and the region displaced is called a shear region. Hence <p takes the following forms in which g if g 2 , gz may- be equal, or any two may be equal: I. [g&SpyQ + g 2 pSya() + g^ySapOVSapy; reduced equations for g x = g 2 or g x = g 2 = # 3 ; II. [ 9l aSPyQ + giPSyaQ + mScfiO + a0Sj8y()]/So0y; reduced equation for gi = g 2 , or if a — 0; III. g + [(a/3 + cy)80yQ + bySya ()]/SaPy, reduced if a = = c, or a = = b = c. EXAMPLES (1). Let <pp=V-app, where SaP + 0. Take X = a, u = P, v = Vafi, then we find with little trouble mi = - Sap, m 2 = - a 2 /? 2 , ra 3 = a 2 p 2 SaP, and the characteristic equation of <p, (tp + Sap)(<p - Tap)(<p + Tap) = 0. Hence there are three invariant lines in general, and oper- ating on p by (<p + Safi)(<p — TaP), we find the invariant line corresponding, (<p + SaP)p = aSpp + pSap, (<p- TaP)(<p + Sap)p = a 2 pSPp + P 2 aSap - aTaPSPp - pTaPSap = - (TaSpp+ TpSap)(Ua+ UP)Tap. Hence the invariant line corresponding to the root TaP is Ua + Up. The other two are Ua - Up and UVap. THE LINEAR VECTOR FUNCTION 229 (2). Let <pp = Vafip. (3). Let <pp = g 2 aS(3yp + frtfSyap + ySo&p) + hfiSaPp. (4). Let <pp= gp+ (fifi + ly)Sfap + r(3Syap. (5). Let <pp = Vep. 6. Solution of cpp = a. It is obvious that when <p satis- fies the general equation <p 3 — mi<p 2 + m 2 (p — ra 3 = 0, ra 3 4= 0, then the vector m%<p~ l p = (w 2 — miv? + <^ 2 )p. For if we take the <p function of this vector, we have an identity for all values of p. Also this vector is unique, for if a vector a had to be added to the left side, or could be added to the left side, then it would have to satisfy the equation <pa = 0. But if ra 3 4= 0, there is no vector satis- fying this equation, for this equation would lead to a zero root for <p. Hence, if cpp = X, ra 3 p = m 2 X — mnp\ + <p 2 X, which solves the equation. If <p satisfies the general equation (pi — mnp 2 + m<2<p = 0, m% #= 0, then we have one and only one zero root of the latent equa- tion, and corresponding to it a unique vector for which <pa = 0, and if (pp = X, m 2 p = xa + [m\(p — (p 2 )p = xa + w&iX — <pX. If (p satisfies the cubic (p z — rriiv 2 =0, mi 4 0, the vacuity is two, and we have two cases according as there is not a reduced equation, or a reduced equation exists 230 VECTOR CALCULUS of the form <p 2 — m\<p = 0. In either case the other root is mi. There is a corresponding invariant line X, and if the vector a is such that <pa = 0, then we have in the two cases a vector (3 such that respectively <p(3 = a, or <p(3 = 0. Hence, if <pp = 7, we must have in the two cases 7 = x\ + yot, or 7 = x\. Otherwise the equation is impossible. Hence mip = x\ + za + yj3 = 7 + ua -f 2/0, where ^>/3 = a, <pa = 0, or where <pfi = = ^ck- If ^> satisfies the cubic and no reduced equation, there are three vectors (of which fi and 7 are not unique) such that <py = fi, <fP = a, <^a = 0, and then <pp = X, we must have X = xa + yft where p is any vector of the form p = za + iCjS + 1/7. If <p 2 = 0, and no lower degree vanishes, then <p(x(3 + 2/7) = <*j ^a = 0, and X = ua. If <p = 0, there is no solution except for <pp = 0, where p may be any vector. 7. Zero Roots. It is evident that if one root is zero, then the region <p\ where X is any vector will give us the other roots. For instance let <pp = Vep. Then if /x = Veh, cpp, = Xe 2 — eSe\, <p 2 fi = e 2 /x, and the other two roots are ± V — 1 • Te. If two roots are zero, then <p 2 on any vector will give the invariant region of the other root. For instance, let THE LINEAR VECTOR FUNCTION 231 <pp = aSfiyp, then aSfiyaSfiyp = <p 2 p. Hence cpa = aSapy gives the other root as Sapy and its invariant line a. In case a root is not zero, but is g\, if it is of multiplicity one, then <p — gi operating upon any vector will give the region of the other root, or roots. If it is of multiplicity two, then we use (<p — g{) 2 on any vector. 8. Transverse. We define now a linear vector operator related to <p, and sometimes equal to <p, which we shall indicate by <p' ', and call the conjugate of <p, or transverse of <p, and define by the equation S\<pijl = Sn<p'\ for all X, /z. For example, if <pp = Vap(3, then S\<pp = S\ap(3 = SpfiXa, and <p' = VPQa = <p, if <pp = Vep, <p'p = — Vep; if <pp = aSfip, <p'p = (3Sap. If a is an invariant line of <p, (pa — ga, then for every /S 8p<pa = gSaP = Sa<p'P, or Satf - g)P = 0, that is a is perpendicular to the region not annulled by <p[ — g, that is invariant for <p' — g. If we consider that from the definition we have equally S\(p 2 p, = Sfxcp' X, S\(p z iJL = Sup' X, it is clear that <p and <p' have the same characteristic equa- tion and the same general equation. They can differ only in their invariant regions if at all. If then the roots are all distinct, it is evident that the invariant line a of <p, is normal to the two invariant lines of <p' corresponding to the other two roots, hence each invariant line of <p is normal to the two of <p' corresponding to the other roots, and conversely. If now the characteristic equation is the general equation, 232 VECTOR CALCULUS so that each function satisfies only the general equation, let there be two equal roots, g, whose shear region gives <pa= ga + ft <p@ = g(3, let <py = giy. Then &Vp = gSap + Sj3p, S/Vp = gSfip, Sycp'p = 0i#yp, Sapy<p'p = g(V(3ySap + VyaSfo) + F)M0P + giVa(3Syp. Therefore corresponding to the root g\, <p' has the in- variant line Vafi, and to the root g, the invariant line V(3y. Further (p f converts Vya into gVya + Vfiy. Hence the invariant line of g\ for <p' is normal to the shear region of g, and the shear region of g for <p f is normal to the invariant line of g\ for <p, but the invariant line of g for ip' is normal further to the shear direction of g for <p, and the shear direction of <p' for g is normal to the invariant line of (p for g. In case there are three equal roots, and no reduced equa- tion, we have <pa = ga + ft <p/3 = gfi + 7, <£>7 = PY, so that &Vp = gSap + Sft>, Wp = gS(3p + S 7 p, #7<p'p = gSyp, Sapy<p'p = p^Sa/fy + VfhfSfo + F7CKS7P. Hence, the invariant line of <p' is Vfiy, its first shear line Vya, and second shear line Vafi. In case there is a reduced equation with two distinct roots, we have <p(xa -f- y&) = 5f(ara + yfi), <P7 = 0i7, Sa<p'p = gSap, S/Vp = gSfip, Sy<p'p = giSyp, Sa&y-<p'p = gVfiySap + gVyaSfip + giVa(3Syp, THE LINEAR VECTOR FUNCTION 233 Hence, the invariant line of <p' corresponding to gi is normal to the invariant plane of g for <p, corresponding to g there is an invariant plane normal to the invariant line of gi for <p. Every line in the plane through the origin is invariant. In case the reduced equation has two equal roots, then <pa = ga + ft <pP = gP, <py = gy, Sa<p'p = gSap + S(3p, Sy<p'p = gSyp, S(3<p'p = gSfip, Sa(3y<p'p = gp + Sfa-iVfa), Corresponding to g, we have then two invariant lines: Va.fi, which is perpendicular to the shear plane of <p; V(3y, which is perpendicular to the non-shear region of g and to the shear direction of g; also the shear direction of <p' is Vfiy, so that the shear region of <p' is determined by Vya and Vfiy, and is therefore perpendicular to y. The three forms of <p' are I. <p' = [giVfySaO + toVyaSpQ + g 3 Va(3Sy01ISapy; II. <p' = faVpySaQ + giVyaSpQ + aVfaSPQ + g 2 VaPSyQ]/Safrr, III. <p' m g + [aVfiySPQ + bVyaSyQ + cVpySyQ]/SaPy. We may summarize these results in the theorem : The invariant regions of ip' corresponding to the distinct roots are normal to the corresponding regions of the other roots for <p. In case there are repeated roots, if there is a plane every line of which through the origin is invariant, then every line of the corresponding plane will also be in- variant, but if there is a plane with an invariant line and a shear direction in it, the first invariant line of the con- jugate will be perpendicular to the shear direction and to the second invariant line of <p, and the shear direction of the conjugate will be perpendicular to the invariant lines of ip; 16 234 VECTOR CALCULUS while finally, if there is an invariant line, a first shear direc- tion, and a second shear direction, then the invariant line of the conjugate mil be perpendicular to the invariant line and the first shear direction of <p, the first shear direction will be perpendicular to the invariant line and the second shear direction of <p, and the second shear direction will be perpendicular to the two shear directions of <p. Let a, /3, y define the various directions a = V(3y/Sa(3y, /? = Vya/Sa/3y, y = VaP/Sa(3y, then we have <p = gioSoi + gtffS0 + gzySy) <p' = giaSa + g 2 @S(3 +. g 3 ySy J or or ( gi aSa + gJISp + afiSa + g 2 ySy) \ gi aSa + gi (3S(3 + aaS(3 + g 2 ySy\ ig+(aJl+cy)Sa+byS(3 \g+aaSp+ (b(3 + ca)Sy . 9. Self Transverse. It is evident now that <p = <p' only when there are no shear regions, if we limit ourselves to real vectors, and further the invariant lines must be per- pendicular or if two are not perpendicular, then every vector in their plane must be an invariant, and even in this case the invariants may be taken perpendicular. Hence every real self-transverse linear vector operator may be reduced to the form <pp = — aSapgi — (3S(3pg 2 — ySypg 3 , where a /3 y form a trirectangular system, and where the roots g may be equal. Conversely, when <p — <p', the roots are real, provided that we have only real vectors in the system, for if a root has the form g + ih, where i — V — 1, then if the invariant THE LINEAR VECTOR FUNCTION 235 line for this root be X + ip,, where X and p are real, we have <p(\ + in) = (g + ih)(k + ifi) = g\ — hp + i(h\ + gp) = <p\-\- iipjJL. Therefore <p\ = g\ — hp, <pp = hX + gfx, and $/*<pX = gS\p — hp 2 = S\<pp = AX 2 + <7$Xju. Thus we must have ^X 2 + hf? = 0. It follows that h = 0. Of course the roots may be real without <p being self- transverse. An important theorem is that <p tp' and <p'<p are self- transverse. For Sp<p(p'(T = Sa<p<p'p, Sp<p'<p<r — S(T(p f (pp. EXERCISE Find expressions for <p<p' and <p'<p in terms of a, /3, 7, a, jS, 7. 10. Chi of p. We define now two very important func- tions related to <p and always derivable from it. First X* = m i — <P> so that Sa(3y-x<pP — pSafi<py + pS(3y<pa + pSyapfi — (paSfiyp — cpfiSyap — <pySa(3p = VVaPV<pyp + • • • = aSp((3<py — y<p(3) + • • •. The other function is indicated by \j/^ or by x vv and defined 4r p = m 2 — mi(p + <p 2 = ra 2 — <px v , Sapy-i/z^p = pSonpfiipy + • • • — <paSp((3(py — y<p(3) = aSp<p(3(py -\~ f3Sp<py<pa + ySp<pa<p(3. 236 VECTOR CALCULUS We have at once from these formulae the following im- portant forms for FX/x, X„FX/x = [aSVlniVpvy - Vy<pp) • • .]/SaPy = [aS(V<p'\n - V\<p'n)V0y + • • -]/SaPy = *W + V\i/>% Whence we have also <pV\p = miV\ix — V\<p'n — Vp'XfjL, 1^FX/x = [aSV\»V<pp<py H ]/Sapy = V<p'\<p'lL. Since it s evident that X+ = x/j and \p v , = #/, we have at once x\V\ii = V<p\» + V\<pfi ^FX/x = V(p\<pji. The two expressions on the right are thus shown to be functions of FX/x. It is evident that as multipliers of p ™<i = <P + X = f' + X'i ^2 - *>X + lA = *>'x' + ^', m 3 = ^ = ^V- EXERCISES 1. If <p = aiSPiQ + a 2 SM) + «*Sfo(), show that <p' = faScHQ + 2 Sa 2 () +01&I.O, X = 27/9i7ai(), * = - 2F/3 1/ 8 2 5Fa l a 2 (), mi = 2*Sau9i, ra 2 = — Z£Faia 2 F(8i/3 2 , ra 3 = — Saia 2 azS0ifi 2 3 , X ' = UFaiV/SiO, ^ = - HVaiatSVPiPiQ. 2. Show that the irrotational invariants of x and ^ are mi(x) = 2m h m 2 (x) = »ii 2 + m 2 , w 3 (x) = Wim 2 — m 3 ; rai(^) = ra 2 , m 2 (^) = raira 3 , Wj(^) = m 3 2 . THE LINEAR VECTOR FUNCTION 237 3. For any linear vector function <p, and its powers <p 2 , <p 3 , • • • , we have mi(<f?) = Wi 2 — 2ra 2 , m 2 (<p 2 ) = m 2 2 — 2raira 3 , w 3 (^) = ra 3 2 . mi(<p 3 ) = mi 3 — 3wiW 2 + 3m 3 , m 2 (<p 3 ) = 3raira 2 ra 3 — m 2 3 — 3ra 3 2 , m 3 (<p 3 ) = m 3 3 . mi(^ 4 ) = mi 3 — 4mi 2 w 2 + 2w 2 2 + 4raim 3 w 2 (<p 4 ) = w 2 4 — 4wiw 2 2 w 3 + 2wi 2 m 3 2 + 4ra 2 ra 3 2 , m 3 (^> 4 ) = ra 3 4 . 4. Show that for the function <? + c, where c is a scalar multiplier, mi(<p + c) = wi(^) + 3c, m 2 ((p + c) = 0ts(?) + 2mi(<p)c -f 3c 2 , wi 3 (¥> + c ) = w s(«p) + cw 2 (<p) + c 2 mi(<p) + c 3 . 5. Study functions of the form x\p + ?/x + 2. 6. <p'V<p\<pfi = m 3 V\n; <p'(V\<pn — F/x^X) = m 2 F\M — V<p\<pn. 7. ^(a^>) = aHiv)', tifilPi) = ^(<Pi)-^('Pi)' 8. «A(a) = a 2 , ^[7a()J = - aSaQ, *(- 0Sa) = 0. +{— QxiSi - gsjSj - g 3 kSk) = - g 2 g 3 iSi — g 3 gijSj - gig 2 kSk. = - VptfiSVaiyi - VPtppSVata, - VptfiSVa&n. 10. For any two operators <p, 9, mi(<pd) = mi(M, m. 2 (<pd) = m 2 (6<p), m 3 (<pd) = tr»»(0*). mi((p6) = mi(<p)mi(0) + ra 2 (<p) + ra 2 (0) — m 2 {6 + <p). m 2 (<pd) = m 2 {6)m 2 (<p) + m 3 {<p)-m v {d) + ra 3 (0)-rai(y?) - mtffto + *'(*)]. m 3 (<pd) = m 3 (<p)-m 3 {6). rrii(<p + 0) = mi{<p) + Wi(0). m 2 (v? + 0) = m 2 (v?) + m 2 (0) + mi(6)'ini(ip) - nii(<pO). mt(<p + 0) = m 3 (*>) + m 3 (0) + mi[*V(4) + 0V(*>)]. 11. x can have the three forms : t (ff* + 9t)*Sa + (g 3 + 0i)0S0 + (oi + g 2 )ySj; II. fo + o 2 )a£5 + fo + fln)/aSg + 2^x7^7 + apSa; III. 2g - (a/3 + c 7 )>S£ - bySfT The operator x is the rotor dyadic of Jaumann. 12. The forms of \f/ for the three types are I. g 2 g g aSa + g 3 gi&Sl3 + gig 2 ySy; II. gig 2 aSZ + g 2 gi0S8 + 0i 2 7#t" - agtfSZ; III. o 2 - [O03 + (ab - gc)y]Sa - bgySfi. 238 VECTOR CALCULUS 13. An operator called the deviator is defined by Schouten,* and is for the three forms as follows: I. (l9i - 9* - gs)aSZ + (Itfi - 0* - gi)0S8 + (lg* - g x - g%)ySy'; II. (- fci - <7*)(«S5 + fiSfi) + (§0» - 2^)7^ + apSZ; III. (o£ + Cy)Sa + bySd. It is V<p = <p — S<p, where S(<p) = \m\. 14. Show that if F (X, M ) - - F (m, X) then F (X, M ) - C (X, m)Q.VX m , where C is symmetric in X, n and Q is a quaternion function of VX/i. 11. We derive from <p and ^' the two functions That there is a vector e satisfying this last equation, and which is invariant, is easily shown. For if we form ™>z(<P — <p')> we find that S(<p - <p')\{<p - <p')n(<p - (p')v = S(p\<piJL<pi> — 2$ (p\<p' )jl<p' v — ^LSipkcpyup'v—Sip'Xv'mp'v = S\iAi>(m 3 — ra 3 + Wi(^» — mi(^')). But it is easy to see that this expression vanishes identically, for the first two terms cancel, and if <p lt <p 2 are any two linear vector functions, we have = Siiv<p\kSiiv<p<Lh + SjjLvcpiiiSvXip^K + SiiV(pivS\mp2K + Sl>\<Pi\SlJlV<p2lJL + Sv\<PillSv\(p2lJL + Sp\<PipS\/JL(P2H + S\fJL<Pi\SlJLV<p2V + S\lJL<PilJ,ST<P2V + $XjU<pi J/jSAjU^ = S^knv - mi(<p2<pi) . Hence we may under mi permute cyclically the vector functions. Again after this has been done we may take the conjugate. Hence the expression above vanishes, and there is a zero root in all cases for <p — <p'. Further we may always write * Grundlagen der Vector- und Affinor-Analysis, p. G4. THE LINEAR VECTOR FUNCTION 239 S\fXV<pp = (pXSfJLPp + • • * S\jJLl>'<p'p = VjivS\<p f p + • • • = VfivSipKp + Hence we have S\nv(<p - <p') P = V P V(Vfip)cp\ + .... From this we have 2eS\pv = V(p\Vpv + • • • for every noncoplanar X, p, v. The function <p is evidently self-transverse, and the conjugate of VeQ is — VeQ. It is easy to show that 2<peS\pv = — V\V<pp(pv — • • •. The expressions Te, T<pe, and Sepe are scalar invariants of <p, and these three may be called the rotational invariants. In terms of them and the other three scalar invariants all scalar invariants of <p or <p' may be expressed. If there are three distinct roots, g\, g 2 , g 3 , and the corre- sponding invariant unit vectors are y h y 2 , 73, we may set these for X, p, v, and thus 2e&7iy 2 Y3 = giVjiVy 2 y3 + g 2 Vy 2 Vy z yi + gzVy z Vy^y 2 = (92 — g3)yiSy 2 y 3 + (g 3 - gi)y2Sy s y 2 + (gi — g2)yzSyiy 2 . 2<peSyiy 2 y 3 = - g 2 g 3 Vy x Vy 2 y z — g 3 g\Vy 2 Vy z y x — gig2Vy 3 Vyiy 2 . In case two roots are equal and (pa = g x a + (3h 2 , <p(3 = #i/3, (py = g 2 y, we have 2eSa(3y = (g 2 - gi)VyVafi + VQVPyh. In case three roots are equal, <pa = ga-\-h(3, <pfi = gr/5+ ly, <py = gy 2eSa(3y = h(3V(3y + lyVya. It is evident, therefore, that if the roots are distinct and 240 VECTOR CALCULUS the axes perpendicular two and two, that « = 0; if two roots are equal and the invariant line of the other root is perpendicular to the plane of the equal roots, then it is the direction of e; and if the three roots are equal, and if the invariant line is perpendicular to the two shear directions, then € is in the plane of the invariant line and the second shear. 12. Vanishing Invariants. The vanishing of the scalar invariants of (p leads to some interesting theorems. If Wi = 0, there is an infinite set of trihedrals which are transformed by <p into trihedrals whose edges are in the faces of the original trihedral. If ^transforms any trihedral in this manner, mi = 0, and there is an infinite set of trihe- drals so transformed. We choose X, n, v for the edges of the vertices, and if <p\ is coplanar with /z, 7, <pix with v, X, and <pv with X, ju, the invariant mi = 0. If mi = 0, we choose X, ju, arbitrarily, and determine v from ScpXnv = = SXcpuv. Then also S\n<pp = 0. The invariant m 2 vanishes if <p transforms a trihedral into another whose faces pass through the edges of the first. The converse holds for any infinity of trihedrals. EXERCISES 1. Show that if a, fi, 7 form a trirectangular system mi = — Sa<pa — S/3<pl3 — Sy<py and is invariant for all trirectangular systems, m 2 (<p<p') = T*<poc + TV/? + T*<py, TV(X) = S 2 \<pa -f £ 2 Xv/3 + S 2 \<py. 2. Study the functions for the ellipsoid and the two hyperboloids - <p = a^aSa ± b~ 2 fiSl3 ± c^ySy. 3 Study the functions ZmVaVQct, <P + VaVQa, a^VoapQ, r-VpVaQ, V -vVaQ.fi. THE LINEAR VECTOR FUNCTION 241 4. Show that V <pp = 2e — mi, \/Sp<pp = — 2 (pop, VAp = — 2<pt — m 2 , \7Vp<pp = 2Sep + Wip — 3<pp, wherein <p is a constant function. Hence (pop may always be repre- sented as a gradient of a scalar, Sep as a convergence of a vector, and m,\p — 3<pp (deviation) as a curl. We may consider also that Wi is a convergence and e is a curl, ra 2 a convergence and <pe a curl. 5. An orthogonal function is defined to be one such that ip<p' = 1. Show that an orthogonal function can be reduced to the form ip = () cos - sin 0-70/3 = (lT cos 9)0800 = 0**l*Q0-*l* or — /3(0/" - )+ 1 ()/? -(fl ./ 7r)-1 which is a rotation about the axis /3 through the angle — 0, or such a rotation followed by reflection in the plane normal to /?. 6. Study the operator <p 112 . 7. Show that m ■i(<po) = m h m 2 (<Po) = mi + e 2 , m 3 (<p ) = m 3 ■ f 5-6V56. Hence if Tf Te = 0, m 2 (<p) = m 2 (<p ). u 8. Show that Se<pt i = 0, m 3 (<p) = m 3 (<p ). mAVe{)] = o, m*[VeQ] = TV, m 3 [Ve()] = 0. 9. Show that e(x) = - ■ 6, «(x) - " * e ' e( ^ _1) = " m 3 ipe. 10, 11, , Show that . if = y./M), \p(<Po) = * + aSe(). rni(d) = 2S/3e, m-i (0) = - S/3*>/3, m 3 (d) = o, " 2 (a 2 — aSa), 12. If *> = F-a(), ,p 2 » = ^2n+l = a 2n7 a () > 13. For any two operators <p, 0, 2eM) = 2e(^ o 0o) + X(<p)e(6) + x{0)e(<p) + V-e(<p)e(6). 242 VECTOR CALCULUS In particular 14. An operator ^> is a similitude when for every unit vector a, T^a = c, a constant. Show that the necessary and sufficient condition is <p'<p = c 2 . Any linear transformation which preserves all angles is a similitude. 15. If <p = aSi + 0Sj + ySk, then <p' = iSa +jSp + kSy, and ^j^' = — ctSa — fiSfi — 7$7, mi(*V) = Pa + P/3 + 7*7, m,(^^') = PFa/S + 7*7/37 + T^a, mz(<p<p') = — S 2 a0y. 13. Derivative Dyadic. There is a dyadic related to a variable vector field of great importance which we will study next. It is called the derivative dyadic, since it is somewhat of the nature of a derivative, as well as of the nature of a dyadic. This linear vector function for the field of a will be indicated by D a and defined by the equation D.= ~ SQV-<r. It is evident at once that if we operate upon dp, we arrive at da. This function is, therefore, the operator which en- ables us to convert the various infinitesimal displacements in the field into the corresponding infinitesimal changes in the field itself. The expression SdpDJp = Cdf, where C is a constant and dt a constant differential, repre- sents an infinitesimal quadric surface, the normals at the ends of the infinitesimal vectors dp being D a dp. Let us consider now the field of a, containing the con- gruence of vector lines of <r. Consider a small volume given by 8p at the point whose vector is p, and let us sup- THE LINEAR VECTOR FUNCTION 243 pose it has been moved to a neighboring position given by the vector lines of the congruence, that is, p becomes p + adt. Then p + 8p becomes p+8p + dt(<r + DM, that is to say, dp has become (1 + Dadt)8p. Hence any area V8\p8 2 p becomes, to terms of the first order only, V8 lP 8 2 p + dt(V8 lP D,8 2 p + VDJ lP 8 2P ). The rate of change with regard to t of the vector area V8ip8 2 p is therefore X (D ff )V8 lP 8 2 p. Likewise, the infinitesimal volume S8ip8 2 p8 s p is trans- formed into the volume S8ip8 2 p8 3 p + dt{S8ip8 2 pD a 8 z p + S8ipD a 8 2 p8 3 p + SD a 8ip8 2 p8 3 p). The rate of increase of the volume is, therefore, miS8ip8 2 p8 3 p. In other words if we displace any portion of the space of the medium so that its points travel infinitesimal distances along the lines of the congruence of a, by amounts propor- tional to the intensity of the field at the various points, then the change in any infinitesimal line in the portion of space moved is given by dtD ff 8p, the change in any infinitesimal area is given by x'(D a )dt- Area, and the change in an infinitesimal volume is midt times the volume. In case a defines a velocity field the changes mentioned will actually take place. We have here evidently a most important operator for the study of hydrodynamics. If adt is the field of an infinitesimal strain, then D a 8p is the 244 VECTOR CALCULUS displacement of the point at dp. Evidently the operator plays an important part in the theory of strain, and con- sequently of stress. Further, (we shall not stop to prove the result as we do not develop it) for any vector a a function of p we have an expansion analogous to Taylor's theorem, in the series h 2 <r(p + ha) = (r(po) + hD^ + -^ (- &*V)Z).a + | (SaV) 2 D a a + •••• This formula is the basis of the study of the singularities of the congruence. For if cr(p ) = 0, then the formula will start with the second term, and the character of the con- gruence will depend upon the roots of D ff . In brief the results of the investigation of Poincare referred to above (p. 38) show that if none of the roots is zero, we have the cases : 1. Roots real and same sign, the singularity is a node. 2. Roots real but not all of the same sign, a faux. 3. One real root of same sign as real part of other two, a focus. 4. One real root of sign opposite the real part of others, a faux-focus. 5. One real root, other two pure imaginaries, a center. If one or more roots vanish, we have special cases to con- sider. The invariants of D a are easily found, and are mi = — SV<r, e = ^Vxja, m 2 = — %SVViV2V<tkt 2 , D*e = iV-VViV2V<ri<T2, m 3 = |SViV2V3<W- 2 0- 3 . After differentiation, the subscripts are all removed. The related functions are THE LINEAR VECTOR FUNCTION 245 BJ = - v&r(), X = - VVV*Q, %' = ~ V-VQV-*, $ = - jrviV2^K7 2 (), y = - i&oviVs-Wi*!. In a strain a the dilatation 's ra b the density of rotation (spin) is e, and in other cases we can interpret m\ and e in terms of the convergence and the curl of the field. In case a is a field of magnetic induction due to extraneous causes, and a is the unit normal of an infinitesimal circuit of electricity, then %'« is the negative of the force density per unit current on the circuit. In any case we might call — x'V8i P 8 2p the force density per unit circuit. Since x' is not usually self-transverse, the force on circuit a has a component in the direction jS different from the component in the direction a of the force on circuit ft. Recurring to Stokes' and Green's theorems we see that fdpa = ff - WVd lP d 2P '(T = 2ffS8 lP 8 2 pe -Sfx'V8 lP 8 2P . It is clear that the circulation in the field of a is always zero unless for some points inside the circuit e is not zero. The torque of the field on the circuit vanishes for any normal which is a zero axis of x'« From these it is clear that, if we have a linear function <pd P , in order that it be an exact differential da we must have the necessary and sufficient conditions VVvO m 0. For if tf<pd P = 0, then <pVUvV = for all Uv, whence the condition. The converse is easy. The invariant m 3 in the case of the points at which a = will be sometimes positive, sometimes negative. A theorem given originally by Kronecker enables us to find what the excess of the number of roots at which ra 3 is positive over the number of roots at which ra 3 is negative is.* We set * Picard, Traite d'Analyse, Vol. I, p. 139. 246 VECTOR CALCULUS t = fa/To* and 7 = - J- ffSdvr] 47T then the integral will vanish for any space containing no roots, and will be the excess in question for any other space. We could sometimes use this theorem to determine the number of singularities in a region of space and something about their character. It is evident that <SVr = 0. The operator (D c ) = \(D + DJ) is called the deforma- tion of the field, and the operator Ve() the rotation of the field. In case a is a unit vector everywhere, then DJa = 0, and since the transverse has a zero root, D a itself must have a zero root. There is one direction then for which D ff a = 0. The vector lines given by Vadp = are the isogons of the field. In case there are two zero roots the isogons are any lines on certain isogon surfaces. EXERCISES 1. Study the fields given by a = — p, a = Up/p 2 , a = Vap, a = aSfip, a = Vap/p 3 . 2. Show that if a is a function of p, a + da = — V o[Spo<r — %Spo<ppo] — \V Po^V <* = VVoihVvpo - Wpo<PPo] - lSV<r, where Vo operates only on p , and <p = — <rSS7 0- The first form expresses a + da as a gradient and a term dependent on the curl of a, the second as a curl and a term dependent on the convergence of a. po is an infinitesimal vector. 3. If a = FVr, D a = ZV. 14. Dyadic Field. If <p is a linear vector operator de- pendent upon p, we say that <p defines a dyadic field. For every point in space there will be a value of <p. Since there is always one root at least for <p which is real, with an in- variant line, there will be for every point in space a direction THE LINEAR VECTOR FUNCTION 247 and a numerical value of the root which gives the real invariant direction and root. These will define a con- gruence of lines and a numerical value along the lines. In case the other axes are also real, and the roots are distinct or practically distinct, there will be two other related con- gruences. The study of the structure of a dyadic field from this point of view will not be entered into here, but it is evidently of considerable importance. EXERCISES 1. If <p = uQ, then the gradient of the field is Vw. The vorticity of the field is VV <p() — VVuQ. The gradient in any case is v'V, a vector. 2. If <p = VaQ, the gradient is — V\7<r, the vorticity is QSS7*+D V m - x {D,r). 3. If <p = <tSt(), the gradient is aSVr — D T <r, the vorticity is WvStQ + V<rD y (). The gradient of the transverse field is tS\7<t - DaT, the vorticity VX/tSvQ + VtD<j{). 4. If ip = VadQ, the gradient is - 70(V)<r + VadV, the vor- ticity is S\7<T-d() +S*V'-0'Q -*S$[(V i)Q +£<r0(). For the transverse field we have the gradient is — 0'FV'o - — 0VV<r, the vorticity 7v W«r() + W'dVa'Q. 5. If <p = D,r the gradient of the field is — VV, the concentration of <r, and the vorticity is D vva • The gradient of the transverse field is — V^Vo", while the vorticity is zero. 6. If <p = VV0(), the gradient is FV0V, where both V's act on 0, and the vorticity is V 2 0() - V&V0(). 7. If <p = De(a), the gradient is — 7^, the vorticity is Dvve<r- 8. If f> = to, the gradient is 2e(07V0). 9. For any <p Vm 1 = < P V +2e (FWO ), Vm 2 = 2 € {<pW<p' + FW' I, Vw 3 = 2 e [V (Vi M *' -t W Vx]. 248 VECTOR CALCULUS 15. The Differentiator. We define the operator — SQ V as the differentiator, and indicate it by D. It may be used upon quaternions, vectors, scalars, or dyadics. As examples we have, D being the transverse B v „ = VaD r () - VtD Q, D Sar - SQD.r + S()D T a, D Vaa = - VaD.Q, D mi M = mriDJ, D eM = e(DJ, D v = -S()V •*>(). 16. Change of Variable. Let F be a function of p, and p a function of three parameters u, v, w. Let A = ad/du + f3d/dv + yd/dw, where a, /3, y form a right-handed system of unit vectors. Then we have the following formulae to pass from expres- sions in terms of p to differential expressions in terms of the parameters. AF = - AiS Pl VF t FA' A" = |FAi'A 2 "£Fpip 2 Fv'V", SA'A"A'" - - i<SAi'A 2 "A8 , "iSpiP2PsSV' V'V". As instances - SVv= A'VV'V, VA<r= VV"T(r"A'. Notations Dyadic products 4>(a), <f>'(a), <f)Va( ), Va(f>( ), Hamilton, Tait, Joly, Shaw. <l>'a f a-4>, <j) X a, aX <j>, Gibbs, Wilson, Jaumann, Jung. Reciprocal dyadic 4>~ l , Hamilton, Tait, Joly, Gibbs, Wilson, Burali-Forti, Marcolongo, Shaw. q~ l , Timerding. I6I" 1 , filie. THE LINEAR VECTOR FUNCTION 249 The adjunct dyadic \j/ = m(f)'~ l , Hamilton, Tait, Joly, Shaw. WO2, Gibbs, Wilson, Macfarlane. R{a), Burali-Forti, Marcolongo. x((f>, (f>), Shaw. D4>~ 1 , Jaumann, Jung. The transverse or conjugate dyadic <f>', Hamilton, Tait, Joly. 0, Taber, Shaw. <f> c , Gibbs, Wilson, Jaumann, Jung, Macfarlane. K(ct), Burali-Forti, Marcolongo. \b / , Elie. The planar dyadic X = Wi — (f> r , Hamilton, Tait, Joly. 4>J — <f> c , Gibbs, Wilson. — </>/, Jaumann, Jung. CK(a), Burali-Forti, Marcolongo. x(0), Shaw. Self-transverse or symmetric part of dyadic <f>o t Hamilton, Tait, Shaw. $, Joly. <f> f , Gibbs, Wilson. [</>], Jaumann, Jung. D(a), Burali-Forti, Marcolongo. \ b /, Elie. \ b° / , Elie. In this case expressed in terms of the axes. Skew part of dyadic \{4> — </>') = V-e( ), Hamilton, Tait, Joly, Shaw. </>", Gibbs, Wilson. II, Jaumann, Jung. Va A , Burali-Forti, Marcolongo. 17 i 250 VECTOR CALCULUS \ b / , £lie. Sin <f>, Macfarlane. Mixed functions of dyadic X«>, 0), Shaw. \<f>l 0, Gibbs, Wilson. R{(f>, 0), Burali-Forti, Marcolongo. Vector of dyadic e, Hamilton, Tait, Joly. <£ x , Gibbs, Wilson. (f> r 8 , — <}>/, Jaumann, Jung. Va, Burali-Forti, Marcolongo. E, Carvallo. R = Te, filie. c(<£), Shaw. Negative vector of adjunct dyadic <f>e, Hamilton, Tait, Joly. 0-0 x , Gibbs, Wilson. <t>-<f> r 8 , Jaumann, Jung. olVol, Burali-Forti, Marcolongo. «x(</>> <f>)> Shaw. Square of pure strain factor of dyadic 4><f>', Hamilton, Tait, Joly. </></> c , Gibbs, Wilson. {(f)} 2 , Jaumann, Jung. aKa, Burali-Forti, Marcolongo. [6], filie. </></>', Shaw. Dyadic function of negative vector of adjunct <f> 2 e, Hamilton, Tait, Joly, Shaw. <f> 2 -4> x , Wilson, Gibbs. THE LINEAR VECTOR FUNCTION 251 2 -0/, Jaumann, Jung. a 2 Va, Burali-Forti, Marcolongo. K 2 , Elie. Scalar invariants of dyadic. Coefficients of characteristic equation m" ', ra', m, Hamilton, Tait, Joly, Carvallo. 1%, h, h, Burali-Forti, Marcolongo, Elie. F, G, H, Timerding. S , (</>2) s , 03, Gibbs, Wilson, mi, ra 2 , ra 3 , Shaw. fc, ] 4> 8 *, >■ • • • 03, Jaumann, Jung. - w, J cos </>••• 03, Macfarlane. (Mer scalar invariants ™>i(<f>o 2 ), mi(00'), 2(rai 2 — m 2 ), rai(00')> wi[x(0, *)> 0L Shaw. [0 8 ] 2 «, {0j s 2 > [01/, •'* -j Jaumann, Jung. • • •, • • •, • • ., : 0, 0* : ft Gibbs, Wilson. Elie uses ifi for $e0e. Notations for Derivatives of Dyadic In these V operates on unless the subscript n indicates otherwise. Gradient of dyadic V0, Tait, Joly, Shaw. Dyadic of gradient. Specific force of field 0V, Tait, Joly, Shaw, grad a, Burali-Forti, Marcolongo. -3 — , Fischer. dr 252 VECTOR CALCULUS Transverse dyadic of gradient 0'V, Tait, Joly. grad Ka, Burali-Forti, Marcolongo. —r^-y Fischer. V -<t>, Jaumann, Jung. Divergence of dyadic - SV<f>( ), Tait, Joly, Shaw. X grad Ka, Burali-Forti, Marcolongo. Vortex of dyadic VV4>( ), Tait, Joly, Shaw. Rot a, Burali-Forti. V X 0, Jaumann, Jung. Directional derivatives of dyadic - S( ) V • 0. Sa' 1 V ■ <l>a. ScT 1 V -<t>Va(), Tait, Joly, Shaw. S(a, ( )), Burali-Forti. P , IX*, F i sch e r . da da Burali-Forti, Marcolongo. (»<>)<»• Gradient of bilinear function ju„(Vn, «), Tait, Joly, Shaw. <£(/z)a, Burali-Forti. Bilinear gradient function ju(Vn, u n ), Tait, Joly, Shaw. \//(n, u), Burali-Forti. Planar derivative of dyadic <f> n VVn( ), Tait, Joly, Shaw. X-^> Fischer, CHAPTER X DEFORMABLE BODIES Strain 1. When a body has its points displaced so that if the vector to a point P is p, we must express the vector to the new position of P, say P', by some function of p, cpp, then we say that the body has been strained. We do not at first need to consider the path of transition of P to P'. If cp is a linear vector function, then we say that the strain is a linear homogeneous strain. We have to put a few restrictions upon the generality of <p, since not every linear vector function can represent a strain. In the first place we notice that solid angles must not be turned into their symmetric angles, so that SipKcpyupvlSKp.v must be positive, that is, ra 3 is positive. Hence (p must have either one or three positive real roots. The corresponding invariant lines are, therefore, not reversed in direction. 2. When <p is self-conjugate there are three real roots and three directions which form a trirectangular system. The strain in this case is called a pure strain. Any linear vector function can be written in the form #rr V.{**9."f I 0« = p -1 V(<pV)()-p, where q-i()q = (wT'V The function <p<p' is self -conjugate and, therefore, has three real roots and its invariant lines perpendicular. If we set 7r = V ((p<p') y then 7r 2 = ipip'. Let the cubic in <p<p' be G 3 - M X G 2 + M 2 G - M 3 = 0. Then from the values given in Chapter IX, p. 237, for the coefficients of <p 2 253 254 VECTOR CALCULUS in terms of those of <p we have (the coefficients of the cubic in w being p u p 2 , p 3 ) Mi = pi* - 2p 2f M 2 = p 2 2 — 2pm, M 3 = p z 2 , whence we have P! 4 - 2(Mi + 8M 3 )pi 2 - \m 2 M zVl + MS - 4M 2 2 M 3 = 0. Thence we have pi, p 2 , and p 3 . Now if the invariant lines of <p<p' are the trirectangular unit vectors a, 0, 7, we may collect the terms of <p in the form <P = aaSa'Q + bpSP'Q + cySy'Q, where a, b, c are the roots of V <p<p' = w and a! , fi', 7' are to be determined. Hence <p' = aa'SaQ + • • • and - tp'ip = tfa'Sa'Q + VP'Sp'Q + WO- But also ^' = _ otefo _ fc0S0 - c 2 7#7, since a, /?, 7 are axes of <p(p', and a 2 , b 2 , c 2 are roots. Now we have <p'a = — act', <p'/3 = — b(3', <p'y = — cy', hence <p(p r a = a 2 a = — a 2 otSa'ct' — ab(3Sa'(3' — acySot'y'. Thus we have a:' 2 = — 1, Set' (3' = = Sa'y', and similar equations, so that a', (3', y' are unit vectors forming a tri- rectangular system, and indeed are the invariant lines of <p'<p. We may now write at once 7r = — aaSct — b/3S(3 — cySy, q ~\) q = - aSa' - fiSfi* - ySy'. This operator obviously rotates the system a', (3', y' into DEFORM ABLE BODIES 255 the system a, (3, y, as a rigid body. That the function is orthogonal is obvious at a glance, since if we multiply it by its conjugate we have for the product - aSa - PSP - ySy = 1(). Reducing it to the standard form of example five, Chapter IX, p. 236, we find that the axis is UV(aa' + (3(3' + 77') and the sine of the angle of rotation \TV{aa! + $8' + 77') • EXAMPLES (1). Let <p = VeQ- Then <p' = - VeQ, <P<P' = ~ VeVeQ = eSe() - e 2 . The axes are e for the root 0, and any two vectors a, j8 perpendicular to e, and these must be taken so that a(3 = Ue, the roots that are equal being T 2 e. We may therefore write <p = TeaS(3 - Te(3Sa = V-eQ, which was obvious anyhow. Hence we have for q~ l Qq the operator aSp-0Sa= V(VaP)Q, and this is a rotation of 90° about Va(3 = Ve of 90°. The effect of 7T = Te(- aSa - (3S(3) is to give the projection of the rotated vector on the plane perpendicular to e, times Te. That is, finally, VeQ rotates p about € as an axis through 90° and annuls the component of the new vector which is parallel to e. (2). Consider the operator g — aS(3Q where a, jS are any vectors. It is to be noticed that we must select of all the square roots of (p<p f that one which has its roots all positive. It is obvious that j) = q. 256 VECTOR CALCULUS 3. The strain converts the sphere Tp= r into the ellip- soid 7V -1 p = r, or WW = - r\ This is called the strain ellipsoid. Its axes are in the direc- tions of the perpendicular system of (p<p' — tt 2 . The ellip- soid Sp<p'<pp = — r 2 is converted into the sphere Tp = r. This is the reciprocal strain ellipsoid. Its axes are in the directions of the principal axes of the strain. The exten- sions of lines drawn in these directions in the state before the strain are stationary, and one of them is thus the maxi- mum, one the minimum extension. 4. A shear is represented by <PP — P ~ fiSap, where Sa/3 = 0. The displacement is parallel to the vector /3 and proportional to its distance from the plane Sap = 0. There is no change in volume since ms = 1. If there is a uniform dilatation and a shear the function is <pp = gp- fiSap. The change in volume is now g 3 . The equation is easily seen to be (<P - 9? = 0. This is the necessary and sufficient condition of a dilatation and a shear, but this equation alone will not give the axes and the shear plane, of course. 5. The function <pp = gqpcT 1 ~ qfiq~ l Sap is a form into which the most general strain can be put which is due to shifting in a fixed direction, U(5, planes parallel to the fixed plane Sap = by an amount proportional to the perpen- dicular distance from the fixed plane, then altering all lines in the ratio g } and superposing a rotation. This is DEFORMABLE BODIES 257 any strain. We simply have to put <p'<p into the form <p'<p = b 2 + X£ju + ftSk, where S\fx = i(a 2 + c 2 - 2b 2 ), T\n = K« 2 - c 2 ), and then we take g =b, a - - X, bp = n- IXrK* ~ c ) 2 - The rotation is determined as before. 6. All the lines in the original body that are lengthened in the same ratio, say g, are parallel to the edges of the cone TcpUp = g or SUp(<p f <p — g 2 )Up = 0, or in terms of X, /z, 2SX UpSfx Up = b 2 - g 2 , sin u-smv= (b 2 - g 2 )/(a 2 - c 2 ) , where u and v are the angles the line makes with the cyclic planes of the cone Scpp<pp = — T 2 p. 7. The displacement of the extremity of p is 5 = a — p = O — l)p, which can be resolved along p and perpendicular to p into the parts p(/Sp~Vp — 1) + pVp~ l <pp. The coefficient of p in the first term is called the elongation. It is numerically the reciprocal of the square of the radius of the elongation quadric: Sp(<p — l)p = — 1, the radius being parallel to p. The other component may be written Vep + Vcpopp~ l - p, where e is the invariant vector of <p, the spin-vector. 8. If now the strain is not homogeneous, we must con- sider it in its infinitesimal character. In this case we have again the formula da = — SdpV -tr = cpdp, where a is now the displacement of P, whose vector is p, and a + da that of 258 VECTOR CALCULUS p -f- dp, provided that we can neglect terms of the second order. If these have to be considered, da = - SdpV a + i(SdpV) VV = (pdp — %SdpV - (pdp. We may analyze the strain in the case of first order into <P = (fo + VeQ. Since now € = \V\7<r, if e = 0, it follows that a = VP and there is a displacement potential and p '« - VSVP(). The strain is in this case a pure strain. If e is not zero, there is rotation, about e as an axis, of amount Te. In any case the function <p determines the changes of length of all lines in the body, the extension e of the short line in the direction Up being — SUppoUp. The six coefficients of <p , of form — Sa<po(3, where a, ft are any two of the three trirectangular vectors a, ft 7, are called the components of strain. Three are extensions and three are shears, an unsymmetrical division. 9. In the case of small strains the volume increase is — S\7<7, and this is called the cubic dilatation. If it vanishes, the strain takes place with no change of volume, that is, with no change of density. A strain of this char- acter is called a transversal strain. There is a vector potential from which a can be derived by the formula a = VVt, SVt = 0. There is no scalar potential since we do not generally have also VVo- = 0. Indeed we have 2 e = VV<r = WVVr = V 2 r - VSVr = V 2 r. DEFORMABLE BODIES 259 This would give us the integral \t = \irfffejr-dv. The integration is over the entire body. This strain is called transverse because in case we have a a function of a single projection of p, on a given line, say a, so that a = af v x + /3f 2 -x + yf 3 -x, SVo- = — /i = 0, fi = constant, and all points are moved in this direction like those of a rigid body. We may therefore take the constant equal to zero, and /i = 0, so that Saa = 0. Hence every displacement is perpendicular to the line a. 10. When V\/a = 0, we call the strain longitudinal; for, giving <j the same expression as in § 9, we see that we have Wa =0 = 7/2' - fifs', and / 2 = = / 3 , Vaa = 0. Hence we have all the strain parallel to a. 11. In case the cubical dilatation iSVo" = 0, the strain is purely of a shearing character, and if the curl VVv = 0, the strain is purely of a dilatational character. Since any vector a can be separated into a solenoidaJ and a lamellar part in an infinity of ways, it is always possible to separate the strain into two parts, one of dilatation only, the other of shear only. If we write a = VP + V\/t, then we can find P and r in one way from the integrals P = lir.ffSS<T'VTp-W, r = - \TT'fffVa'VTp- l -dv f , p = p' - Pc . 260 VECTOR CALCULUS The integrations extend throughout the body displaced. This method of resolution is not always successful, and other formulae must be used. (Duhem, Jour, des Math., 1900.) 12. The components are not functionally independent, but are subject to a set of relations due to Saint Venant. These relations are obvious in the quaternion form, equiva- lent to six scalar equations. The equation is V-V<PoVV() = 0, if <p=SQV-<r, where both V's operate on <p . The equation is, further- more, the necessary and sufficient condition that any linear vector function <p can represent a strain. The problem of finding the vector a when <p is a given linear and vector function of p consists in inverting the equation <p = — S() V -cr. (Kirchkoff, Mechanik, Vorlesung 27.) It is evident that if we operate upon dp, we have <pdp = do. Hence the problem reduces to the integration of a set of differential equations of the ordinary type. EXAMPLES (1). If (p = VeQ, we have or = Vep. Prove Saint Venant's equations. (2). If <p~ p- l V{)p-\ then a = Up. Prove Saint Ven- ant's equations. 13. In general when we do not have small strains, we must modify the. preceding theory somewhat. The dis- placement will change the differential element dp into dpi = dp — SdpV-<r. The strain is characterized when we know the ratio of the two differential elements and this we may find by squaring DEFORMABLE BODIES 261 so as to arrive at the tensor (dpi)* = Sdp[l - 2vSa + V'S(r'(T"SV"]dp- The function in the brackets is the general strain function, which we will represent by <£. It is easily clear that if <p = — SQV'<r then * = (1 + <p)(l + <p') = (1 + *>)(1 + <p)'. Of course $ is self-conjugate. Its components Sa&fi are also called components of strain. If <p is infinitesimal, we may substitute (1 + 2<po) for <£. The cubical dilatation is now found by subtracting 1 from SdipidhpidtPi/Sdipdtpdtp = m 3 (l + <p) = 1 + A. Evidently (1 + A) 2 = m 3 ($). The alteration in the angle of two elements is found from - suq. + <p)\u(i<p)y. If angles are not altered between the infinitesimal elements, the transformation is conformal, or isogonal. In such case Eti&k' = s 2 \ys\$\s\'$y. For example, if <p = VaQ, sua + <p)\Q. + <p)v = sxx', when Sa\ = = Sa\'. 14. This part of the subject leads us into the theory of infinitesimal transformations, and is too extensive to be treated here. On Discontinuities 15. If the function <j is continuous throughout a body, it may happen that its convergence or its curl may be dis- continuous. The consideration of such discontinuities is 262 VECTOR CALCULUS usually given at length in a discussion of the potential functions. Here we need only the elements of the theory. We make use of the following general theorem from analysis. Lemma. If a function is continuous on one side of a sur- face for all points not actually on the surface in question, and if, as we approach the surface by each and every path leading up to a point P, the gradient of the function, or its directional derivatives approach one and the same limit for all the paths; then the differential of this function along a path lying on the surface is also given by the usual formula, — SdpV -q = dq, dp being on the surface. [Hadamard, Lecons sur la propagation des ondes, etc., p. 84, Painleve, Ann. Ecole Normale, 1887, Part 1, ch. 2, no. 2.] In the case of a vector a which has the same value on each side of a surface, which is the value on the surface, and is the limiting value as the surface is approached, at all points of the surface, we have on one side of the surface da — — Sdp\7 •& = <pidp. On the opposite side da = — Sdp\7 -<r = <p2dp. If now these two do not agree, but there is a discontinuity in <p, so that <p 2 — tp\ is finite as the two paths are made to approach the surface, then designating the fluctuation or saltus of a function by the notation [], we have in the limit [da] = (<p 2 — <Pi)dp = [<p]dp. But since a does not vary abruptly, [da] along the surface is zero, hence for dp on the surface [<p]dp = 0, DEFORMABLE BODIES 263 and therefore M = — vSv> where v is the unit normal, \x a given vector. That is to say, we have for the transition of the surface [S()V-a] = »Sv. Whence [SVcr] = Spix, [W<t] = Vvix. These are conditions of compatibility of the surface of dis- continuities and the discontinuity; or identical conditions, under which the discontinuities can actually have the sur- face for their distribution. 16. If *S/x^ = 0, then [S Vo"] = 0, and the cubic dilatation is continuous. Since Svvjjl = = Sv[V\7<t] = [SpV<t], the normal com- ponent of the curl of a is continuous, and the discontinuity is confined to the tangential component. Likewise Sfivn = = [S/xVo-], and the component along ijl is continuous. Hence V\7(r can be discontinuous only normal to the plane of /jl, v. 17. In case a itself is discontinuous, the normal com- ponent of a as it passes the surface of discontinuity cannot be discontinuous without tearing the surface in two. Hence the discontinuity is purely tangential. It can be related to the curl of a as follows. . . Consider a line on the surface, of infinitesimal length, and an infinitesimal rectangle normal to the surface, and let the value of a at the two upper points differ only infinites- imally, as likewise at the two lower points, but the differ- ence at the two right hand points or at the two left hand 264 VECTOR CALCULUS . points be finite, so that a has a discontinuity in going through the surface equal to [a]. Then fSbpa = ffSK(AWa) around the rectangle, when k is normal to the rectangle. But the four parts on the left for the four sides give simply Sid*}, where 8p is a horizontal side and equal to V-wTSp. Hence we have for every k tangential to the surface SkV v[a] - Sk Urn (AW<r)IT8p. Dropping all infinitesimals, we have Vv[(t] = Lim AVVcr/Tdp. Tangential discontinuities may therefore be considered to be representable by a limiting value of the curl multi- plied by an infinitesimal area, as if the surface of discon- tinuity were the locus of the axial lines of an infinity of small rotations which enable one space to roll upon the other. The expression \[<j] is the strength of this sheet. A strain is not irrotational unless such surfaces of dis- continuity are absent. But we have shown above that a continuous strain may imply certain surfaces of discon- tinuity in its derivatives of some order. If V\7cr = 0, everywhere, then Vv[u] = 0, and such discontinuity as exists is parallel to v. The derivation above applies to any case, and we may say that if a field is irrotational, any discontinuity it pos- sesses must be normal to the surface of discontinuity. Integrating in the same way over the surface of a small box, we would have ffSv[<r]ds = SV<T'V, DEFORMABLE BODIES 265 where v is the infinitesimal volume. But this gives Sv[a] = vSV (r/surface. If then $Vo" = everywhere, the discontinuity of a is normal to the normal, that is, it is purely tangential. These theorems will be useful in the study of electro-dynamics. Kinematics of Displacements 18. In the case of a continuous displacement which takes place in time we have as the vector a the velocity of a moving particle, and if p is the vector from a fixed point to the particle, then dp/dt = a. It is necessary to distin- guish between the velocity of the particle and the local velocity of the stream of particles as they pass a given fixed point in the absolute space which is supposed to be sta- tionary. The latter is designated by d/dt. Thus dcr/dt is the local rate of change of the velocity at a certain point. While da/dt is the rate of change of the velocity as we follow the particle. It is easy to see that for any quaternion q the actual time rate of change is dq/dt = dq/dt — SaV -q. We have thus the acceleration da/dt = da/dt - SaV-cr = (d/dt + <p)a. If the infinitesimal vector dp is considered to be displaced, we have bdp/dt = - S5pV'(r. Since the rotation is \V\7a dt, the angular velocity of turn of the particle to which dp is attached is |FVo". This is the vortex velocity. Likewise the velocity of cubic dilata- tion is — S\/a. The rate of change of an infinitesimal volume dv as it 18 266 VECTOR CALCULUS moves along is — SV<T'dv. The equation of continuity is d(cdv) = 0, where c is the density, or dc/dt + c{- SV<r) = 0. That is, we have for a medium of constant mass dc/dt = cSVv- That is, the density at a moving point has a rate of change per second equal to the density times the convergence of the velocity. It may also be written easily dc/dt = SVW. This means that at a fixed point the velocity of increase in density is equal to the convergence of the momentum per cubic centimeter. 19. When FVo" = 0, the motion is irrotational, or dila- tational, and we may put a = VP, where now P is a veloc- ity-potential, which may be monodromic or polydromic. When SVcr = 0, the motion is solenoidal or circuital, and we may write a = VVr where &Vr = 0. r is the vector potential of velocity. The lines e = \V\7<r become in this case the concentration of Jr. The lines of a are the vortex lines of r, and the lines of e are the vortex lines of a. 20. If a is continuous, and the equation of a surface of discontinuity of the gradient dyadic of a and of a' is / = 0, where now a is a displacement and a' is da/dt the velocity, we have certain conditions of kinematic compatibility. These were given by Christoffel in 1877-8 and are found as follows. We have M = o, [_0ov«W<-jtfi> DEFORMABLE BODIES 267 in the case in which the time t is not involved; and for a moving surface in which / is a function of t as well as of p, we would have [-SOV-<r]=-»SUvfO, [" S Tt V<T ] = " mS i Uvf= M f /*V/=M=-Gm. This gives us the discontinuity in the time rate of change of the displacement of a point as it passes from one side to the other of the moving surface. The equation of the surface as it moves being /(/>, t), we have in the normal direction - SdpV-f+dtf = 0, that is, since dp is now Uvfdn, dn/dt — — f'/T\/f = G, where / ' is the derivative of / as to t alone. In words, at any point on the instantaneous position of the moving surface the rate of outward motion of the point of the surface coinciding with the fixed point in space is G = —f'jTS/f. The moving surface of discontinuity is called a wave and G the rate of propagation of the wave at the given point. We may now read the condition of com- patibility above in these words: the abrupt change in the displacement velocity is given by a definite vector p. at each point multiplied by the negative rate of propagation of the wave of displacement, that is, if G is the rate of propagation, [o-'j = - Gp, and [SVff] = - SpVvf = - S/iv. 21. The preceding theorem becomes general for discon- tinuities of any order in the following way. Let the func- tion a and all its derivatives be continuous down to the (n — l)th, then we can write [SQiV'SQzV — S0*-iV-*]«0, 268 VECTOR CALCULUS whence, differentiating along the surface of discontinuity as before, we find in precisely the same manner • [S()iV • • • SO. V •*) = nSOiUvfSihUvf • • • SQnUvf, since at a given point on the fixed surface V/ is constant. And if we insert dp/dt in m parentheses (m < ri), we shall have, since the surface is moving, = - »G»SQiUvf • • • S0n-mUvf(-l) m . In particular for m — 2 = n, we have W) = mG 2 , which is the discontinuity in the acceleration of the dis- placement. If m = 1, n = 2, [SOW] = - nGSQUVf. From this we derive easily [SVff'l = - GS»Uvf= - GSfxp. [W<r'] = - GVfxUvf= ~ GVfip. 22. The nth. derivatives of Saa are [S()iV • • • SQnV-Saa] = SQiUvf • • ■ SQ n UVfSap. If then we hold the surface fixed and consider a certain point, the discontinuity in the nth derivative of the ratio of two values of the infinitesimal volume which has two perpendicular directions on the surface and the third along . the normal will be given by the formula SQiUvf ■ ■ ■ SOnllVfSnUvf. In case we have a material substance that has mass and DEFORMABLE BODIES 269 density and of which the mass remains fixed, we have c/cq = volo/voi, log c — log Co = log v — log V, V log c = — V log v/v = — Vo/v-V(clvo). Therefore from the formula above we have since v /v = 1 in the limit [SOiV • • • S()nV log c] = SQiUVf • • • SQnUvfSfjiUvf. In particular for the case of discontinuities of order two- we have [Vlogc]= UvfSfiUvf. 23. These theorems may be extended to the case in which the medium is in motion as well as the wave of discontinuity. Stress 24. In any body the stress at a given point is given as a tension or a pressure which is exerted from some source across an infinitesimal area situated at the point. The stress real y consists of two opposing actions, being taken as positive if a tension, negative if a pressure. It is as- sumed that the stress taken all over the surface of an infinitesimal closed solid in the body will be a system of forces in equilibrium, to terms of the first order. This is equivalent to assuming that the stress on any infinitesimal portion of the surface is a linear function of the normal, that is 6 = ZVv. 25. We have therefore for any infinitesimal portion of space inside the body ffQdA = ffZdv = 0. But by Green's theorem this is equal to the integral through 270 VECTOR CALCULUS the infinitesimal space J J VHV = 0. Hence SV = 0. In this equation S is a function of p, and V differentiates S. 26. In case the portion of space integrated over or through is not infinitesimal, this equation (in which S is no longer a constant function) remains true if there is equilibrium; and if there are external forces that produce equilibrium, say £ per unit volume, then the density being c, we have SV + c£ = for every point. In case there is a small motion, we have EV + c£ = co". 27. Returning to the infinitesimal space considered, we see that the moment as to the origin of the stress on a portion of the boundary will be VpSJJv and the total moment which must vanish, considering S as constant, is ffVpZdv = fffVpttdv, hence FpHv = = €(S). We see therefore that S is self-conjugate. EXAMPLES (1). Purely normal stress, hydrostatic stress. In this case S is of the form pS = gp, where g is + for tension, — for pressure, and is a function of p (scalar, of course). (2). Simple tension or pressure. H = — paSa. (3). Shearing stress. H = - p(aSp + PSa), |S not parallel to a. DEFORMABLE BODIES 271 (4). Plane stress. 8 - giaSa + g 2 (3S(3. (5). Maxwell's electrostatic stress. H= l/87r-FvP()VP, where P is the potential. 28. The quadric Spap = — C is called the stress quadric. Its principal axes give the direction of the principal stresses. Since Sp is the direction of the normal we may arrive at a graphical understanding of the stress by passing planes through the center, and to each construct the conjugate diameter. This will give the direction of the stress, and since Tap is inversely proportional to the perpendicular from the origin on the tangent plane at p, if we lay off on the conjugate diameter distances inversely as the per- pendiculars, we shall have the vector representation of the stress. When the diameter is normal to its conjugate plane, there will be no component of the corresponding vector that is parallel to the plane, that is, no tangential stress. Such planes will be the principal planes of the stress. It is evident that a stress is completely known when the self-conjugate linear vector function H is known, which depends therefore upon six parameters. We shall speak, then, of the stress H, since H represents it. This proposi- tion is sometimes stated as follows: stress is not a vector but a dyadic (tensor). From this point of view the six components of the stress are taken as the coordinates of a vector in six-dimensional space. These components in the quaternion notation are, for a, (3, y, a trirectangular system, - SaXa, - S/3E0, - SyZy, - Saafi = - S(3Za, - SpEy = - SyZp, - SyZa = - Sa3y. 272 VECTOR CALCULUS That is, X x Y y Z t , Xy = Y X i Y t = Zy, Zi x ~ X 2. It is easy to see now that certain combinations of these component stresses are invariant. Thus we have at once the three invariants mi, m 2 , m 3 , which are X x ~r* Yy~\~ %zt YyZz -f- ZgX x ~r X x Y y — Y z — Z x — X y , X t Y yZ z -\- ZXyY Z Z X X X Y z Y y Z x Z z X y . For any three perpendicular planes these are invariant. EXERCISE What are the principal stresses and principal planes of the five ex- amples given above? 29. Returning to the equation of a small displacement, we may write it er" = i + <T l EV. Hence the time rate of storage or dissipation of energy is W'=- fffSa'Zvdv. The other terms of the kinetic energy are not due to storage of energy. Now we have an experimental law due to Hooke which in its full statement is to the effect that the stress dyadic is a linear function of the strain dyadic. The latter was shown to be <Po= -^S()V-<7+ V&rOJ. The law of Hooke then amounts to saying that S is a linear function of a and V where V operates upon a, and owing to the self-conjugate character of <p, we must be able to interchange V and a, that is, S = 6[(), V, a}. DEFORMABLE BODIES 273 First, it follows that if the strain <p is multiplied by a variable parameter x, that the stress will be multiplied by the same parameter. We have then for a parametric change of this kind which we may suppose to take place in a alone a' = ax' . Hence for a gradually increasing a, we would have W = - xx'fffSaSVdv, w = - iyyy&rEv &% if x runs from to 1. This gives an expression for the energy if it is stored in this special manner. If the work is a function of the strain alone and not dependent upon the way in which it is brought about, W is called an energy- function. It is thus seen to be a quadratic function of the strain. In case there is an energy function, we have for two strain functions due to the displacements cr lf a 2 Si = e[(), en, Vi], H 2 = G[(), o- 2 , v 2 ]- The stored energy for the two displacements must be the same either way we arrange the displacements, hence we have So- 2 e 3 [V3, *i» Vi] = (Scr 1 e 4 [V*i <r 2 , V 2 ], where the subscripts 3, 4 merely indicate upon what V acts. This is equivalent to saying that so far as vector function is concerned, in the form SaG[(3, 7, 5] we can interchange a, (3 and y, 5. Since S is self-conjugate, is self-conjugate, and we can interchange a and (3. From the nature of the strain function we can interchange y, 8. Of course, in the forms above we cannot interchange the effect of the differentiations. 274 VECTOR CALCULUS We have in this way arrived at six linear vector functions <P\l <f22 <P32 <f23 <fn <Pl2> wherein we can interchange the subscripts, and where <Pn = 0[Q,a,a] ••• ^23= 6[(),ft7] v\, a /3 7 being a trirectangular system of unit vectors. We have further a system of thirty-six constituents Cmu c n i2, • • • where Cim = — Sa<pn<x, C1112 = — Sa<pn<x, • • •, each of the six functions having six constituents. These are the 36 elastic constants. If there is an energy function, they reduce in number to only 21, for we must be able to interchange the first pair of numbers with the last pair. There are thus left 3 forms emu 6 of em%, 3 of Cim, 3 of C1212, 3 of C2311, 3 of 02m. In theories of elasticity based upon a molecular theory and action at a distance six other relations are added to these reducing the number of elastic constants to 15. These relations are equivalent to an interchange of the second and third subscript in each form, thus Cim = Ci2is- These are usually called Cauchy's relations, but are not commonly used. (See Love, Elasticity, Chap. III.) Remembering the strain function <p , we can interpret these coefficients with no difficulty, for we have — SaipoCXj • fty, the stress dyadic due to the strain component — Sa&oaj, where a;, a ; are any two of the three a, (3, y. cijki is the component of the stress across a plane normal to otj in the direction a t due to the strain component — Sak<Poai- DEFORMABLE BODIES 275 EXAMPLES (1). If Sij = — Soti<pocxj, show that we have for the energy function W = ^CnnSn + 2cii22SnS 2 2 + i^c 12 i 2 s 12 2 + 201223^12^23 + SCni2*ll*l2 + 2Cii 2S S n S 23 . (2). When there is a plane of symmetry, say in the direc- tion normal to 7, all constants that involve 7 an odd number of times vanish, for the solid is unchanged by reflection in this plane. Only thirteen remain. If there are two per- pendicular planes of symmetry, normal to (3, y, the only constants left are of the types ClUli C1122, Ci212j the plane normal to a is thus a plane of symmetry also. There are nine constants. This last case is that of tesseral crystals. (3). If the constants are not altered by a change of a into — a, (3 into — (3, as by rotation about 7 through a straight angle, then the plane normal to 7 is a plane of symmetry. (4). Discuss the effect of rotation about 7 through other angles. (5). When the energy function exists we have 0(X, fi, v) - 90*, X, v) = - VvQV\\x, where 6' = 6. 30. A body is said to be isotropic as to elasticity when the elastic constants are not dependent upon directions in the body. In such case the energy function is invariant under orthogonal transformation. It must, therefore, be a function of the three invariants of <po, i»i, ra 2 , m 3 . The last is of third degree, while the energy function is a quadratic and therefore can be only of the form W = - Pmi + Am? + Bm 2 . 276 VECTOR CALCULUS P is zero except for gases and is then positive. The con- stant A refers to resistance to compression, and is positive. B is a constant belonging to solids. The form given the quadratic terms by Helmholtz is Am x 2 + Bm 2 = iHm l 2 + £C[2mi 2 - 6m 2 ]. The [] is the sum of the squares of the differences of the latent roots of <po. The constant H refers to changes of volume without change of form, and in such change it is the whole energy, for if there is no change of form, the roots are all equal and the other term is zero. C refers to changes of form without change of volume, since it vanishes if the roots are equal and is the whole energy if there is no cubical expansion m\. For perfect fluids C = 0. The form given by Kirchoff is Km^tpo 2 ) + Kdrm 2 . From which we have B-C = 2KB, 3C = 2K, H= 2K(d + |), C = \K. We may write for solids, liquids, and gases W = Rdm? + Kmifao*) - Pm x . Later notation gives 2K6 = X, K = /x, that is, W = |Xmi 2 + iirriiicpo 2 ) — Pm\. The constants X, \x are the two independent constants of isotropic bodies. We now have for the stress function in terms of the strain function S = Xrai -f- 2/i^o. EXAMPLES (1). In the case of a simple dilatation we know S = p DEFORMABLE BODIES 277 and we have for <po <Po= - JOSOV-ap + ASapQ) = a(). Substituting in the equation above, we have ()p m X(3o) + 2 M o(). The cubical dilatation is thus 3a = p/(X + |m) = p/», where A: is called the modulus of cubical compression. (2). For a simple shear <p, = - a/2-[aSPQ + g&xOL ™i = 0> S = - a/z[«<Sj8() + 0&*()]. If the tangential stress is T, then T = a/j,. M is the shear modulus or simple rigidity. (3). If a prism of any form is subject to tension T uniform over its plane ends, and no lateral traction, we have S = - afSaQ - Xm + 2n<p . From this equation, taking the first scalar invariant of both sides, T = 3mA + 2muh so that rrn= T/(3\+2fi). Substituting, we have 2/i v 2ju(3X +2ju) We write now E = /x(3X + 2/x)/(X + /x)> the quotient of a simple longitudinal tension by the stretch produced, and called Young's modulus. Also we set s = X/(2X + 2/x), Poisson's ratio, 278 VECTOR CALCULUS the ratio of the lateral contraction to the longitudinal stretch. It is clear that if any two of the three moduli are known, the other may be found. We have X = E/[(l + *)(1 - 2*), M - \Ej(X + *), k - IE/(1 - 2s). In terms of E and s we have t»i(S)'« po -m* E (4). If | < s, k < 0, and the material would expand under pressure. If s < — 1, W would not be positive. (5). If Cauchy's relations hold, s = \ and X = /x. For numerical values of the moduli see texts such as Love, Elasticity. 31. Bodies that are not isotropic are called aelotropic. For discussion of the cases and definitions of the moduli, see texts on elasticity. 32. There is still the problem of finding a from cp after the latter has been found from S. This problem we can solve as follows: <t = <tq-\- fp^da = (To — J£<rS Vdp, where V acts on a = o-o + fgW&P ~ hVdpVVv] = *o + fgWdp ~ WiPi ~ p)VVd<r -d-V( Pl - p)VV<r] = <to- Wifii ~ Po)VVao+ f P P Modp -iVQ>i-p)VvM = <ro- \Vifii - Po)VV<ro + f P S l [<Podp - V( Pl - p)W<Po'dp]. We are thus able to express a at any point pi in terms of the DEFORMABLE BODIES 279 values at p of cr, VVc, and the values along the path of integration of <p and FV^oO- EXAMPLES (1). Let us consider a cylinder or prism which is vertical with horizontal ends, the upper being cemented to a hori- zontal plane. Then we have the value of % = — gcySypSyQ, y vertical unit, where the origin is at the center of the lower base. The conditions of equilibrium are S V + c£ =0, or c{ * - gey, J = - gy. That is, the condition is realizable by a cylinder hanging under its own weight. The tension at the top surface is gel where I is the length. Solving for the strain, we have Let a = gcs/E, b = gc(l + s)IE, and note that FWoO = - aVy() - bVyySyQ = - aVy(). The integral is thus °"o — hV(fti — p )e + fp'oiaSyp-dp + bySypSydp + aV( Pl - p)Vydp] = (To— \V(p\ — po)e + Jl?[aSyp-8p + bySypSydp +« VpiVydp — adpSyp + aySpdp] = a — W{pi — p )e + HbyS 2 yp + aV Pl Vyp + haypX, the differential being exact. This gives us as the value of a at pi, 280 VECTOR CALCULUS *l « f + V(pi - p )(ieo + aVypo) + iaFprypi -f \byS 2 yp> f > « constants. Substituting a and 6, and constructing <Po = - J[S()V-«r+ V&r()], we easily verify. If the cylinder does not rotate, we may omit the second term and if the upper base does not move laterally, then the vector f reduces to — ^gcP/E-y, and we have ' = - hgcP/E-y + gcs/2E-Vpyp + gc(l + s)/2E-yS 2 py. A plane cross-section of the cylinder is distorted into a paraboloid of revolution about the axis and the sections shrink laterally by distances proportional to their distances from the free end. (2). If a cylinder of length 21 is immersed in a fluid of density c', its own density being c, the upper end fixed, p the pressure of the fluid at the center of gravity, then we have the stress given by H = - (p + gc'Syp) - g(c - c')(l - Syp)ySy, whence calculating <p , we have <p = 1/E-l- (p + gc'Syp)(- 1 + 2*) - gs(c - c') X (1 - Syp)] - ySy[g(c - c')(l - Syp)l + s)]/E. And a = f + Vdp + p[(- 1 + 2*)p - ^/(c - cO. - Spyg[ce - s(c + c')]/E + 7lh(c ~ c')(l + s)(l - Syp) 2 ' + hgp 2 W-s(c+c')]/E. (3). What does the preceding reduce to if c = a'? Solve also directly. DEFORMABLE BODIES 281 (4). If a circular bar has its axis parallel to y, and the only stress is a traction at each end, equivalent to couples of moment \ira*pt, about the axis of y, a being the radius, that is, a round bar held twisted by opposing couples, we have S = - lidfySnO + VpySyQ), <Po= - HiySpyO + VpySyQ], a = tVpySyp. Any section is turned in its own plane through the angle — tSyp. t is the angular twist per centimeter. (5). The next example is of considerable importance, as it is that of a bar bent by couples. The equations are g = - E/R-Sap-ySyQ, Po - - (1 + s)/R-Sap-ySyQ ~ s/R-Sap-Q, a = iR-i-al&yp + sS 2 ap - sS 2 yap] + sR~ 1 yaS(3pSap — R~ 1 ySapSyp. If the body is a cylinder or prism of any shape with the axis y horizontal, there is no body force nor traction on the perimeter. The resultant traction across any section is ff- EjR-SapdA, which will equal zero if the origin is on the line of centroids of the sections in the normal state, that is, the neutral axis. Thus the bar is stressed only by the tractions. at its terminal sections, the traction across any section being equivalent to a couple. The couple becomes one with axis (3 = ya and value EI/R, where 7 is the moment of inertia about an axis through the centroid parallel to (3. The line of centroids is displaced according to the law - Saa = iR-'S'-yp, 19 2N2 VECTOR CALCULUS so that it is approximately the arc of a circle of radius R. The strain-energy function is \ER~ 2 -S 2 ap, and the potential energy per unit length %EI/R 2 . For further discussion see Love, p. 127 et seq. (6). When E = - E-Syp-OQ, where dy = 0, and 6 = 0', and a may not be a unit vector, show that <Po = ~ (1 + 8)Syp-6Q + sSyp-mi(0), a = (1 + 8)tiSp6p - OpSpy] + mi«[- \yp 2 + pSpy]. See Love, pp. 129-130. 33. We recur now to the equation of equilibrium EV + cf - 0. In this we substitute the value of H = Xmi + 2/^o = - XSVo- - (o-/S() V + V&r()), whence XV*SVcr + m W + n\/SS7<r - cf = 0, or (X + M ) VSVo- + M V 2 c - c? = 0, or equally since VV = VSVa + VVVa, (X + 2 M ) V»SVcr + fiVVVcr - c£ = 0. This is the equation of equilibrium when the displacement and the force £ are given. In the case of small motion we insert on the right side instead of 0, — ca". The traction across a plane of normal v is — (X + iJ,)vSVcr — pV\Jvv, where v is constant. Operating on the equilibrium equa- tion by *SV(), we see that (X+2/z)V 2 SV<r-oSv£= 0. DEFORMABLE BODIES 283 If then there are no body forces £ or if the forces £ are derivable from a force-function P and V 2 P = throughout the body, we see that SVa is a harmonic function. Since rai(E) = Skmi, we see that mi(H) is also harmonic. Again we have (X + m)V#Vo- = - M VV, whence we can construct the operators (X + /xj V£v()£V<r - - mV 2 V&j - - M vvsv(). and adding the two, 2(X + M)VSvSV(r() - - mV 2 (^V() + V&r()) Now we have g = - \SVct - m(^V() + V-ScrO), and since S\7<r is harmonic V 2 H = - /xV 2 (^V() + V&r()) = 2(X + /*) ViS ViSVcrO 2(X + M ) 3& or V#V£V<7() = (1 + s^VSvSVtrQ. V 2 H = ^- ViSvifiO. This relation is due to Beltrami, R. A. L. R., (5) 1 (1892). EXAMPLE Maxwell's stress system cannot occur in a solid body which is isotropic, free from the action of body forces, and slightly strained from a state of no stress, since we have -Wil(E) = 1/8tt-(vP) 2 , 284 VECTOR CALCULUS which is not harmonic. (Minchin Statics, 3d ed. (1886), vol. 12, ch. 18.) 34. We consider now the problem of vibrations of a solid under no body forces, the body being either isotropic or aeolotropic. The equation of vibrations is c<r" = 6( V, V, <r), where S = 6[(), V, <r] as before, and a is a function of both t and p. If the vector co represents the direction and the magnitude of the wave-front, the equation of a plane-wave will be u = t — Sp/co, since this represents a variable plane moving along its own normal with velocity w. By definition of a wave-front the displacement from the mean position is at any instant the same at every point. That is, a is a function of u and t, hence Vo" = — VSp/ooda/du = uT^a/du, and any homogeneous function of V as/(V) gives /V-<r = f{oT l )d n (rldu n , where n is the degree of /. The equation above for wave-motion then is cv" = e[oj-\ or 1 , d 2 a/du 2 ]. If the wave is permanent, a involves t only through u and if the vibration is harmonic of frequency p, <r" = du 2 a/d 2 = - fa. Therefore e[Uu, Uw, a] = ctrT*u. Hence for a plane wave propagated in the direction Uoj DEFORMABLE BODIES 285 the vibration is parallel to one of the invariant lines of the function e[U<a, Uco, ()]. The velocity is the square root of the quotient of the latent root corresponding, by the density. There may be three plane-polarized waves propagated in the same direction with different velocities. The wave- velocity surface is determined by the equation S[e(w-\ co" 1 , a) - ca][e(u-\ co" 1 , (3 - cjSHeC&T 1 , co" 1 , y] = 0, that is, by the cubic of Q[Uu, Uu, ()]. If there is an energy function, Q[Uu, Uu, ()] is self- conjugate as may easily be seen. In such case the invariant lines are perpendicular, that is, the three directions of vibration, U 6 2 , 03, for any direction of propagation are mutually trirectangular. Since W is essentially positive, the roots are positive, and there are thus three real velocities in any direction. If g is a repeated root, there is an invariant plane of indeterminate lines and the condition for such is V[e(«T\ to" 1 , a) - ca][e(^-\ co" 1 , 0) - cfi] = 0, a and /3 arbitrary. There is a finite number of solutions to this vector equation, giving co, and these give Hamilton's internal conical refraction. The vectors terminate at double points of the wave-velocity surface. The index-surface of MacCullagh, that is, Hamilton's wave-slowness surface, is given by 5[0(p, p, a) - ca][G(p, p, (3) - cj8][0(p, p, 7) ~ ey] = 0, a, jS, 7 arbitrary, which is the inverse of the wave-velocity surface, p is the current vector of the surface, just as co for the other surface, the equation being formed by setting 286 VECTOR CALCULUS p = — a> -1 . The wave-surface, or surface of ray- velocity, is the envelope of Sp/o) = 1, or Spp = — 1, where /x = — w _1 . The condition is that given by the equations of the two other surfaces. It is the reciprocal of the index surface with respect to the unit sphere p 2 = — 1, or the envelope of the plane wave-fronts in unit time after passing the origin, or the wave of the vibration propagated from the origin in unit time. The vectors p that satisfy its equation are in magnitude and direction the ray- velocities. When there is an energy function, this ray-velocity is found easily, as follows: The wave-surface is the result of eliminating between 0(/x, p, a) = ca, Q(dp, p, a) + 0(ju, dp, a) + 0(ju, /x, da) = cdcr, Sup = - 1-Spdfi= 0. From the second equation 2SdfxG(<T, a, n) + SdaOiii, fi, a) = cSadX, or by the equations Sdp.e(<r, a, /x) = 0. Hence as dfi is perpendicular to p, we have G(<r, <r, p) = xp. Operate by Sp and substitute the value of x, Q(U<t, Ua, p) = cp. This equation with 6(p,p, a) = ca gives all the relations between the three vectors. See Joly, p. 247 et seq. CHAPTER XI HYDRODYNAMICS 1. Liquids and gases may be considered under the com- mon name of fluids. By definition, a perfect fluid as dis- tinguished from a viscous fluid has the property that its state of stress in motion or when stationary can be con- sidered to be an operator which has three equal roots and all lines invariant, thus E = -p(), where p is positive, that is, a pressure, or S = —p. If the density is c, we have, when there are external forces and motion, the fundamental equation of hydrodynamics <r" = J - c~ l Vp. In the case of viscous fluids we have to return to the general equation c (*" - {) « - Vp - (X + m) VSV o- -mW. 2. When there is equilibrium Vp = c£. If the external forces may be derived from a force function, P, we have Vp = cVP, hence — SdpVp = — ScdpVP, or dp = cdP for all directions. That is, any infinitesimal variation of the pressure is equal to the density into the infinitesimal variation of the force function. In order that there may be equilibrium under the forces that reduce to £, we must have £ subject to a condition, for from Vp = c£, we have V 2 p = Vc£ + cV£, whence ££V£ = 0, and VV% = F£Vlogc. 287 288 VECTOR CALCULUS If £ = VP, the condition is, of course, satisfied, and from the last equation we see that £ is parallel to Vc, that is to say, £ is normal to the isopycnic surface at the point, or the levels of the force function are the isopycnic surfaces. The equation Vp = c£ states that £ is also a normal of the isobaric surfaces. In other words, in equilibrium the iso- baric surfaces, the isopycnic surfaces, and the isosteric sur- faces are geometrically the same. However, it is to be noted that if a set of levels be drawn for any one of the three so that the values of the function represented differ for the levels by a unit, that is, if unit sheets are constructed, then the levels in the one case may not agree with the levels in the other two cases in distribution. The fundamental equation above may be read in words: the pressure gradient is the force per unit volume. Specific volume times pressure gradient is the force per unit mass. We can also translate the differential statement into words thus: the mean specific volume in an isobaric unit sheet is the number of equipotential unit sheets that are in- cluded in the isobaric unit sheet. The average density in an equipotential unit sheet is the number of isobaric unit sheets enclosed. Since dp and dP are exact differentials, we have : Under statical conditions the line integral of the force of pressure per unit mass as well as the line integral of the force from the force function per unit volume are independent of the path of integration and thus depend only on the end points. 3. There is for every fluid a characteristic equation which states a relation between the pressure, the density, and a third variable which in the case of a gas may be the tempera- ture, or in the case of a liquid like the sea, the salinity. Thus the law of Gay-Lussac-Mariotte for a gas is p = const -c (1+ t^t T) f° r constant volume. HYDRODYNAMICS 289 The characteristic equation usually appears in the form pa = RT, where in this case a is the specific volume, the equation reading dP = adp. From this we have dP = RTdp/p. If T is connected with p by any law such as that given above, we can substitute its value and integrate at once. Or if T is connected with the force function P by an equa- tion, we can integrate at once. Example. In the case of gravity and the atmosphere, suppose that the temperature decreases uniformly with the equi- potentials. Since we must in this case take P so that VP will be negative, we have dP = - RTdp/p, T = T - bP, whence dP = -dT/b, dT/T = Rbdp/p, T = T (p/p ) bR . Or again dP/(T -bP)= - R dp/p, 1 - bP/To = (p/po) R . We thus have the full solution of the problem, the initial conditions being for mean sea-level, and in terms of a or c as follows : T= T (p/p ) bR , a= a (p/po) bR -\ p = b-iT [i- (p/p n T= T (l - bTo-'P), c= c (l - To-'bP) »" 1 »" 1 - 1 , p = Pod- To-'bP)*- 1 *- 1 . Absolute zero would then be reached at a height where the 200 VECTOR CALCULUS gravity potential would be P = To/b, and substituting we find c = 0, p = 0. If b is negative, the fictive limit of the atmosphere is below sea-level. For values of bR from oo to 1, for the latter value b = 0.00348 (that is, a temperature drop of 3.48° C. per 100 dynamic meters of height), we have unstable equilibrium, since from the equations above for c we have increasing density upwards. The case bR = 1 is extreme; however, it is mathematically interesting from the simplicity that re- sults. Pressure and temperature would decrease uniformly and we should have a homogeneous atmosphere. This condition is unstable and the slightest displacement would continue indefinitely. Values of bR less than 1 lead still to unstable equilibrium, the state of indifferent equilibrium occurring when the adiabatic cooling of an upward moving mass of air brings its temperature to that of the new levels. For dry air this occurs for bR = 0.2884 = (1.4053 - 1) /1.4053, or a fall of 1.0048° C. per dynamic hectom- eter. See Bjerknes, Dynamic Meteorology and Hydrography. 4. The equation when there is not equilibrium gives us aVp — £ * — a". Let £ = VP, and operate by V*V (), then WaVp = - VV<r". If we multiply by SUv and integrate over any surface nor- mal to Up, we have SfSUvWaVp = - ffSUvW" = - fSdpa". The right-hand side is the circulation of the acceleration or force per unit mass around any loop, the left-hand side HYDRODYNAMICS 291 is the surface integral of WaVp over the area enclosed. If then we suppose that in a drawing we represent the iso- bars as lines, and the isosterics also as lines that cut these, drawing a line for the level that bounds a unit sheet in each case (and noticing that in equilibrium the lines do not in- tersect), we shall have a set of curvilinear parallelograms representing tubes. The circulation of the force per unit mass around any boundary will then be the number of parallelograms enclosed. It is to be noticed that the areas must be counted positively and negatively, that is, the number of tubes must be taken positive or negative, ac- cording to whether Vfl, Vp, the two gradients, make a positive or a negative angle with each other in the order as written. This circulation of the force per unit mass may be taken as a measure of the departure from equilibrium. In the same way we find that if we draw the equipotentials and the isopycnics, we shall have the number (algebraically considered) of unit tubes in any area equal to the circula- tion of the force per unit volume around the bounding curve. If we choose as boundary, for example, a vertical line, an isobaric curve, a downward vertical, and an isobaric curve, the number of isobaric-isosteric tubes enclosed gives the differ- ence between the excess up one vertical of the cubic meters per ton at the upper isobar over that at the lower isobar and the corresponding excess for the other vertical. If the lines are two verticals and two equipotentials, the number of isopotential-isopycnic tubes is the difference of the two excesses of pressure at the lower levels over pressure at the upper levels. These are the circulations around the bound- aries of the forces per unit mass or unit volume as the case may be. 5. If we integrate the pressure over a closed space inside 292 VECTOR CALCULUS the fluid, we have ffyUvdA = fffVpdv = fffc&v. But this latter integral is the total force on the volume enclosed. This is Archimedes' principle, usually related to a body immersed in water, in which case the statement is that the resultant of all the pressure of the water upon the immersed body is equal to the weight of the water dis- placed. If we were to consider the resultant moment of the normal pressures and the external forces, we would arrive at an analogous statement. The field of force, how- ever, need not be that due to gravity. EXERCISE. Consider the case of a field in which there is the vertical force due to gravity and a horizontal force due to centrif- ugal force of rotation. 6. We turn our attention now to moving fluids. A small space containing fluid with one of its points at po may be followed as it moves with the fluid, always con- taining the same particles. It will usually be deformed in shape. The position p of the particle initially at p will be a function of p and of t, say p = (p , t). The particle initially at p + dp will at the same time t arrive at the position p + dip = 6 (p + dp, t) = p — SdipVo-p, hence dip becomes at time t — SdipVo'P = <pdipo. It follows that the area Vdipd 2 p = V(pdip (pd 2 po, and the HYDRODYNAMICS 293 volume — Sdipdtpdzp = — S(pdipo(pd2po<pd s po = — Sdip Q d2Pod d p ' m s ((p) . If the fluid has a constant mass, then we must have cdv = Codvo, or cra 3 = c . This is the equation of continuity in the Lagrangian form. The reference of the motion to the time and the initial con- figuration is usually called reference to the Lagrangian variables. 7. Since dp = — SdpVp = — S<pdp Vp = — Sdpo<p'Vp = — SdpoVoP, VoP = <p'Vp = - VoSpV-p. But the equations of motion are already given in the form aVp = £ - p", hence in terms of the variables po and t we have aVop = <p'(p — p")- This equation, the characteristic equation of the fluid F(p, c, T) - 0, and the equation of continuity, give us five scalar equations expressing six numbers in terms of p and t. In order to make any problem definite then, we must introduce a further hypothesis. The two that are the most common are (1) The temperature is constant, if T is temperature, or the salinity is constant, if T is salinity. In case both variables come in, we must have two corresponding hypoth- eses: 21)4 VECTOR CALCULUS (2) The fluid is a gas subject to adiabatic change. The relation of pressure to density in this case is usually written p = kc y . y is the ratio of specific heat under constant pressure to that under constant volume, as for example, for compressed air, 7 - 1-408. 8. In the integrations we are obliged to pay attention to two kinds of conditions, those due to the initial values of the space occupied by the fluid at t = 0, the pressure p and density c , or specific volume a , at each point of the fluid, and the initial velocities of the particles p ' at p . The other conditions are the boundary conditions during the movement. As for example, consider a fluid enclosed in a tank or in a pipe or conduit. The velocity in the latter case must be tangent to the walls. If we have the general case of a moving boundary for the fluid, then its equation would be /(P, t) = o. If then p' is the velocity, we must have - Sd P Vf+ (df/dt)dt = 0, or - £p'V/+ df/dt = 0. If there is a free surface, then the pressure here must be constant, as the pressure of the air. In order to have various combinations of these conditions coexistent, it is necessary sometimes to introduce discontinuities. 9. If we were in a balloon in perfect equilibrium moving along with one and the same mass of air, the barograph would register the varying pressures on this mass, the ther- mograph the varying temperatures, and if there were a velocitymeter, it would register the varying velocity of the mass. From these records one could determine graphically or numerically the rates of change of all these quantities as HYDRODYNAMICS 295 they inhere in the same mass. That is, we would have the values of dp/dt, dT/dt, dp/dt. These may be called the individual time-derivatives of the quantities. As the balloon passed any fixed station the readings of all the instruments would be the same as instru- ments at the fixed stations. But the rates of change would differ. The rates of change of these quantities at the same station would be for a fixed p and a variable t, and could be called the local time-derivatives, or partial derivatives. They can be calculated from the registered readings. The relation between the two is given by the equation d/dt = d/dt - Sp'V. Thus we have between the individual and the local values the relations The last equation gives us the individual acceleration in terms of the local acceleration and the velocity. From the fundamental equation we have ovp = f - dp' let + w \p' = i - d p'l dt ~ *<j>% where the function 0=-S()V-p', 0'=-VV(), 0o = K-sovy-wo), 2 e = FVp'. This statement of the motion in terms of the coordinates of 296 VECTOR CALCULUS any point and the time is the statement in terms of Eulers variables. Since near po, p = po + po'dt, we have the former function <p at this point in the form <p= - S()Vo-p = l + <ft(- S()V-p') = 1 + d^atpo. Whence m 3 (<p) = 1 + dtmi(6) = 1 + dt{- SVp'). Since the initial point is any point, this equation holds for any point and we have the equation of continuity in the form c - cdtSVp' = c = c + dt-dc/dt(l ~ dtSVp'), or, dropping terms of second order, dc/dt - cSVp' = 0. This is the* equation of continuity in the Euler form. If we use local values, dc/dt- SV(cp') = 0. That is, the local rate of change of the density is the con- vergence of specific momentum. It is obvious that if the fluid is incompressible, that is, if the density is constant, then the velocity is solenoidal. If the specific volume at a local station is constant, then the specific momentum is solenoidal. If the medium is incompressible and homo- geneous, then both velocity and specific momentum are solenoidal vectors. It is clear also that in any case the normal component of velocity must be continuous through any surface, but specific momentum need not be. If any boundary is stationary, then both velocity and specific momentum are tangential to it. HYDRODYNAMICS 297 In the atmosphere, which is compressible, specific mo- mentum is solenoidal, but in the incompressible hydro- sphere, both velocity and specific momentum are solenoidal. Of course the specific volume of the air changes at a station, but only slowly, so that the approximate statement made is close enough for meteorological purposes. If at any given instant we draw at every point a vector in the direction of the velocity, these vectors will determine the vector lines of the velocity which are called lines of flow. These lines are not made up of the same particles and if we were to mark a given set of particles at any time, say by coloring them blue, then the configuration of the blue particles would change from instant to instant as they moved along. The trajectory of a blue particle is a stream line. If the particles that pass a given point are all colored red, then we would have a red line as a line of flow, only when the condition of the motion is that called stationary. In this case the line through the red particles would be the streamline through the point. If the motion is not sta- tionary, then after a time the red particles would form a red filament that would be tangled up with several stream lines. 10. In the case of meteorological observations the di- rection of the wind is taken at several stations simultane- ously and by the anemometer its intensity is given. These data give us the means of drawing on a chart suitably pre- pared the lines of flow at the given time of day and the curves showing the points of equal-intensity of the wind velocity. Of course, the velocity is usually only the hori- zontal velocity and the vertical velocity must be inferred. One of the items needed in meteorological and other studies is the amount of material transported. If the spe- cific momentum in a horizontal direction is cp r , and lines 20 298 VECTOR CALCULUS of flow be drawn, then for a vertical height dz and a width between lines of flow equal to dn, we will have the trans- port equal to Tp'dndz. Since, however, we have for prac- tical purposes dz = — dp, we can write this in the form transport = Tp'dn(— dp). In order to do this graphically we first draw the lines of flow and the intensity curves. An arbitrary outer bound- ary curve is then divided into intervals of arc such that the projection of an interval perpendicular to the nearest lines of flow multiplied by the value of Tp' is a constant. Through these points a new set of lines of flow is constructed. The transport between these lines is then known horizon- tally for a constant pressure drop, by drawing the intensity curves that represent Tp'dn, and if these are at unit values of the transport, they will divide th£ lines of flow into quad- rilaterals such that the amount of air transported horizon- tally decreases or increases by units, and thus the vertical transport must respectively increase or decrease by units, through a sheet whose upper and lower surfaces have pies- sure difference equal to dp = — 1. Towards a center of convergence the lines of flow approach indefinitely close. dn decreases and it is clear that the vertical transport up- ward increases. There may be small areas of descending motion, however, even near such centers. In this manner we may arrive at a conception of the actual movement of the air. Since the specific momentum is solenoidal, we can as- certain its rate of change vertically from horizontal data. For = SVcp' = — dZ/dz + horizontal convergence, or dZ/dz = horizontal convergence of specific momentum. HYDRODYNAMICS 299 Substituting the value of dz, we have dZ/ (— dp) = horizontal convergence of velocity, dZ/dp = dT P 'lds+ Tp'b. where ds runs along the lines of flow, and 5 is the diver- gence per unit ds of two lines of width apart equal to 1. These considerations enable us to arrive at the complete kinematic diagnosis of the condition of the air. On this is based the prognostications. 11. When the density c is a function of the pressure p, and the forces and the velocities can be expressed as gradi- ents, then'we have a very simple general case. Thus let c = f(p), i = V«(p, 0i p' = Vv(p, t), and set Q = u — fa&p, then VQ = £ — aVp, the equations of motion are dp'/dt + 0(p') = VQ, or since p' = Vv, V[dv/dt + iT 2 Vv- Q] = 0. Hence the expression in brackets is independent of p and depends only on t and we have dv/dt+iFW- Q = h(t). We could, however, have used for v any function differing from v only by a function of t, thus we may absorb the func- tion of the right into v and set the right side equal to zero. We thus have the equations of motion dv/dt + JPVfl - Q = 0, dc/dt - SV(cVv) = 0, c = /(p). From these we have v, c, p in terms of p and /. 12. In the case of a permanent motion, the tubes of flow are permanent. If we can set £ = Vw(p), then we place 300 VECTOR CALCULUS Q = u — fadp, and noticing that p' and Q do not depend on t, we have Sp'V-p' = - VQ. If we operate by — Sdp = — S(dsUp'), we have (kSUp'Tp'VTp' on the left, since Sp'V-Up' = 0. Hence from this equation we have at once - SdpGTV - Q) = 0. Hence along a tube of flow of infinitesimal cross-section tiy-4-a This is called Bernoulli's theorem. C is a function of the two parameters that determine the infinitesimal line of flow. Hence along the same tube of flow J(IV - TW) = Q ~ Qo = u - u - f p * adp. In the case of a liquid a is constant and we can integrate at once, giving }ZV- u+ap= C. From this we can find the velocity when the pressure is given or the pressure when the velocity is given. Since the pressure must be positive, it is evident that the velocity square ^ 2{u + C), or else the liquid will separate. This fact is made use of in certain air pumps. In the case of no force but gravity we have u = gz, iTV-g*+ap= C. This is the fundamental equation of hydraulics. We can- not enter upon the further consideration of it here. Vortices. 13. In the case of p' = Vv it is evident that VVp f = 0. When this vector, or the vector e (§9) does not vanish, HYDRODYNAMICS 301 there is not a velocity potential and vortices are said to exist in the fluid. It is obvious that if a particle of the fluid be considered to change its shape as it moves, then e is the instantaneous velocity of rotation. At any instant all the vortices will form a vector field whose lines have the differential equation VdpWp' = = SdpV' p - VSp'dp; that is, Q'dp = dp', or 0'p' = dp'jdt, from which p' . */><%'. These vector lines are called the vortex lines of the fluid. Occasionally the vortex lines may be closed, but as a rule the solutions of such a differential equation as the above do not form closed lines, in which case they may terminate on the walls of the containing vessel, or they may wind about indefinitely. The integral of this equation will usually contain t, and the vortices then vary with the time, but in a stationary motion they will depend only upon the point under consideration. 14. The equations of motion may be expressed in terms of the vortex as follows, since we have and thus Vp'VVp' ' = Sp'V-p'-iVp' 2 ,. Sp'V-p' =2Vp'e + iVp'\ aVp = i - dp' Idt + JVp /2 + 2Vp'e. 15. When now £ = \/u{p, t), and c = f(p), we set P = fadp, giving VP = aVp, and thence VP = Vu - dp' Idt + JVp /2 - 2Vep'. 302 VECTOR CALCULUS Or, if we set II = u-\- Jp' 2 — P, we have dp'/dt + 2Ve P ' = VII. Operate on this with V-V(), and since VV dp'/dt = 2de/dt, and WVep' = SeV -p' - eSVp' - Sp'V-e, de/dt — Sp'V-e = de/dt, SVp' by the continuity equation is equal to c~ l dc/dt = — a~ l da/dt, we have d(ae)/dt = - S(ae)V-p' = 6(ae). This equation is due to Helmholtz. If we remember the Lagrangian variables, it is clear that 6 is a function of the initial vector p and of t, hence the integral of this equation will take the form ae = e fm 'a,e Q = e' ~ s ^^' dt a e = ^(t)a e . But the operator is proved below to be equal to <p itself, so that when £ = Vu, ae = a Se Vo-p = + ao<p€ , or finally we have, if we follow the stream line of a particle, which was implied in the integration above, Cauchy's form of the integral (a/a )e = — *Se Vo'jP, where p is a function of p and t. It is evident now if for any particle e is ever zero, that is, e = 0, that always e = 0. This is equivalent to Lagrange's theorem that if for any group of particles of the fluid we have a velocity potential, then that group will always possess a velocity potential. (It is to be noted that velocity potential and vortex are phenomena that belong to the particles and the stream lines, and not to the points of space and the lines of flow.) It must be remembered too, that this result was on the supposition that the density was a function of HYDRODYNAMICS 303 the pressure alone, and that the external forces £ were conservative. 16. We may deduce the equation above as follows, which reproduces in vector form the essential features of Cauchy's demonstration. (Appell, Traite de Mec. Ill, p. 332.) Let dp/dt = a, and Q = u — fadp, then, remembering that Q is a function of p and t, and p is a function of p and /, da/dt = VQ(p, t). Also VoQ(po, t) = - VoSpVQ = — VoSpda/dt, where Vo operates on p only; or we can write VoQ = <p' da/dt. Hence, operating with FVo( ), we have V\7o(p f da/dt = = d/dt(VVo<p'a). Thus the parenthesis equals its initial value, that is, since the initial value of cp'a is a , and since Vo = <p'V, VVo<p'<r = 2e = V<p'Vv'<r = m z {<p)<p~ l VS7a = 2m 3 (p~ 1 e. Thus we have at once m 3 e = (pe . This is the same as the other form, since ra 3 = a/a Q . This equation shows the kinematical character of e, and that no forces can set up e or destroy it. 17. The circulation at a given instant of the velocity along any loop is I = - fSdpp'. The time derivative of this is dl/dt = tf^SdpS/Sp'p' - Sdpp") = £(- SdpW tip' 2 - Q] ). But this is an inte- gral of an exact differential and vanishes. Hence if the forces are conservative and the density depends on the pressure, the circulation around any path does not change as the particles of the path describe their stream lines. The 304 VECTOR CALCULUS circulation is an integral invariant. This theorem is due to Lagrange. If we express the circulation in the form I - ' - ffSdvVp' = - 2ffSdpe, we see that the circulation is twice the flux of the vortex through the loop. Hence as the circulation is constant, the flux of the vortex through the surface does not vary in time, if the surface is bounded by the stream loop. The flux of the vortex through any loop at a given instant is the vortex strength of the surface enclosed by the loop. If a closed surface is drawn in the fluid, the flux through it is zero, since the vortex is a solenoidal vector. 18. If we take as our closed surface a space bounded by a vortex tube and two sections of the tube, since the surface integral over the walls of the tube is zero, it follows that the flux of the vortex through one section inwards equals that over the other section outwards. Combining these theorems, it is evident that the vortex strength, or wr- ticity, of a vortex tube is constant. Thus the collection of particles that make up the vortex tube is invariant in time. In a perfect fluid a vortex tube is indestructible, and one could not be generated. 19. It is evident from what precedes that a vortex tube cannot terminate in the fluid but must end either at a wall or a surface of discontinuity, or be a closed tube with or without knots, or it may wind around infinitely in the fluid. If a vortex tube is taken with infinitesimal cross-section, it is called a vortex filament. 20. We consider next the problem of determining the velocity when the vortex is known. That is, given e, to find a = p'. We consider first the case of an incompressible fluid, in which the velocity is solenoidal, that is, SVcr = 0. This with the equations at the boundaries gives us the HYDRODYNAMICS 305 following problem : to find a when 2e = FVo", SVer = 0, SUva = at the boundaries, or if infinite a a = 0. This problem has a unique solution, if the containing vessel is simply connected. We cannot enter extensively into it, for it involves the theory of potential functions, and may be reduced to integral equations. However, since SVv = 0, we may set a = VVr, where *$Vr = 0, whence V 2 r = 2e, and we may suppose r is known, in the form T = h7ffSfe/T(p- Po )dv. If we operate upon this by FV( ), we find a formula for a, a = H,2ir-fffVe(p - p )/T\p - Po )dv. As we see, this formula is capable of being stated thus: the velocity is connected with its vortex in the same way as a magnetic field is connected with the electric current density that produces it, the vortex filament taking the place of the cm rent, the strength of current being Tej2ir, and the elements of length of the tube acting like the ele- ments of current. This solution holds throughout the entire fluid, even at points outside the space that is actually in motion with a vortex. Since the equation of the surface of the tube can be written in the form F( P , t) = 0, this surface will move in time. Its velocity of displace- ment is defined like that of any discontinuity, as UvFdF/dt. On one side the velocity is irrotational, on the other it is vortical. On the irrotational side we have the velocity of the form a — V?, and we must have on 306 VECTOR CALCULUS that side the same velocity of displacement in the form UpSUpVP. The energy involved in a vortex on account of the velocity in the particles is K * - \cfffp' 2 dv = " hcfffSp'Vrdv = ¥fff [SV(p'r) - 2Sre]dv = hcffSdvp'r - cfffSredv = — cj J 'fSredv over all space = c/2T.SffSSSSee'lT(p - p )dvdv'. This is the same formula as that of the energy of two cur- rents. In the expression every filament must be considered with regard to every other filament and itself. Examples. (1). Let there be first a straight voitex fila- ment terminating at the top and bottom of the fluid. Let all the motion be parallel to the horizontal bottom. Then Sya = 0, Vye = 0, de/dt = 0. We have then a = VyVw, 2e = — yV 2 w = 2zy, say, w = — 7r l ffz log rdA. For a single vortex filament of cross-section dA and strength k = zdA, we have iv = — k/w log r = — kjir log V (# 2 + 2/ 2 ) a= Vy(p- po)IT>(p- p ).k/T, where p is measured parallel to the bottom. The velocity is tangent to the circles of motion and in- versely as the distance from the vortex filament. The motion is irrotational save at the filament itself. HYDRODYNAMICS 307 For the effect of vortices upon each other, and their relative motions, see Webster, Dynamics, p. 518 et seq. (2). For the case of a vortex ring or a number of vortex rings with the same axis, see Appell, Traite, vol. Ill, p. 431 et seq. 21. In the more general case in which the fluid is com- pressible we must resort to the theorem that any vector can be decomposed into a solenoidal part and a lamellar part and these may then be found. The extra term in the electromagnetic analogy would then be due to a perma- nent distribution of magnetism as well as that arising fiom the current. EXERCISES 1. If Sea = 0, then it is necessary and sufficient that a = M\/P, M being a function of p. 2. Discuss the case Vae = 0. Beltrami, Rend. R. 1st. Lomb. (2) 22, fasc. 2. 3. Discuss Clebsch's transformation in which we decompose <r thus, o- = Vm + lVV. Show that the vortex lines are the intersections of the surfaces I and v, and that the lines of flow form with the vortex lines an orthogonal system only when the surfaces I, u, v are triply orthog- onal. 4. Discuss the problem of sources and sinks. 5. Consider the problem of multiply-connected surfaces, containing fluids. 22. It will be remembered that Helmholtz's theorem was for the case in which the impressed forces had a poten- tial and the density was a function of the pressure. In this case we will have the equation da/dt + 2Vea = { - aVp + JVtf 2 . Operate by |FV( ) and notice that de/dt - eSVa - SaV-e = a - l d(ae)/dt, whence we have the generalized form a- l d(ae)ldt + SeV -<r = iVV£ - fFVaVp. 308 VECTOR CALCULUS If now at the instant t the particle does not rotate and if a is a function of p alone, then at this instant de/dt = JFV£, and the paiticle will acquire an instantaneous increase of its zero vortex equal to the vortex of the impressed force. That is, £ must be peimanently equal to zero if there is to be no rotation at any time. If FV£ = but a is not a function of p alone, then we have a- 1 d(ae)/dt + SeV -<r = - §WaVp. The right side is a vector in the direction of the intersection of the isobaric and the isosteric surfaces. Now if we take an infinitesimal length along the vortex tube, I, the cross- section being A, the vorticity is ATe = m, the mass is cAl = constant = M. Then we have, since ae = AlejM = mlUejM, - SeV-<r = md(lUe)dtaM « - ~ fUeV* - ^^- e l f I I at a- 1 d(ae)/dt+ SeV -a = dmldt-lUe/aM'+ md{We)la Mdt - md(lUe)/dtaM = dm/dt-lUe/aM = Ve-dTe/dt = ±| number of tubes. Hence the moment m of the vortex will usually change with the time unless the surfaces coincide. Thus a rotat- ing particle may gain or lose in vorticity. If then the isobaric and isosteric surfaces under the influence of heat conditions intersect, vortices will be created along the lines of intersections of the surfaces and these will persist until the surfaces intersect again, save so far as viscosity interfeies. 23. Finally we consider the conditions that must be put upon surfaces of discontinuity, in this case of the first order in <r, that is, a wave of acceleration. HYDRODYNAMICS 309 Let c be a function of p only. Then a\/p = dp/dc \7log c, and the equation of motion becomes p" = J — dp/dc • V log c. Let the equation of the surface of discontinuity be f(p , t) = 0, the normal v. Let £, a, p, and c be continuous as well as dp/dc, but p" = a' be discontinuous at the suiface. Then on the two sides of the surface we have the jump, by p. 263, \p"\ = - dp/dc[V log c], or G 2 ix= dp/dc -UVfSfiUVf. It follows, therefore, that we must have V/iUVf = and G = V (dp/dc), or else we have G = and SnUVf = 0. In the first case the discontinuity is longitudinal, in the second transversal. This is Hugoniot's theorem. In full it is: In a compressible but non- viscous fluid there are possible only two waves of discontinuity of the second order; a longitudinal wave propagated with a velocity equal to V (dp/dc), and a transversal wave which is not propagated at all. The formula for the velocity in the first case is due to Laplace. Also we have for the longitudinal waves [&Vo"] = — GSfxUVf, for transversal waves equal to zero. On the other hand, for longitudinal waves, [FVo\| = 0, for transversal, = GVUVf^. 310 VECTOR CALCULUS REFERENCES. 1. Mathematische Schriften (Ed. Gerhart). Berlin, 1850. Bd. II, Abt. 1, p. 20. 2. On a new species of imaginary quantities connected with a theory of quaternions. Proc. Royal Irish Academy, 2 (1843), pp. 424-434. 3. Die lineale Ausdehnungslehre. Leipzig, 1844. 4. Gow: History of Greek Mathematics, p. 78. 5. Ars Magna, Nuremberg, 1545, Chap. 37; Opera 4, Lyon, 1663, p. 286. 6. Algebra. Bologna, 1572, pp. 293-4. 7. Om Directiones analytiske Betejning. Read 1797. Nye Samm- lung af det kongelige Danske Videnskabernes Selskabs Skrifter, (2) 5 (1799), pp. 469-518. Trans. 1897. Essai sur la representation de la direction, Copenhagen. 8. Essai sur une maniere de repr6senter les quantites imaginaires dans les constructions g6ometriques. Paris, 1806. 9. Theoria residuorum biquadraticum, commentation secunde. 1831. 10. Annales Math, pures et appliqu6es. 4 (1814-4), p. 231. 11. Theory of algebraic couples, etc. Trans. Royal Irish Acad., 17 (1837), p. 293. 12. Ueber Functionen von Vectorgrossen welche selbst wieder Vector- grossen sind. Math. Annalen, 43 (1893), pp. 197-215. 13. Grundlagen der Vektor-und Affinor- Analysis. Leipzig, 1914. \\. Lectures on Quaternions. Preface. Dublin, 1853. 15. Note on William R. Hamilton's place in the history of abstract group theory. Bibliotheca Mathematica, (3) 11 (1911), pp. 314-5. 16. Leipzig, 1827. 17. Leipzig. 18. Elements of Vector Analysis (1881-4), New Haven. Vol. 2, Scientific Papers. INDEX. Acceleration 27 Action : 14, 28 Activity 15, 129, 142 Activity-density 15, 131 Algebraic couple 4, 65 Algebraic multiplication 9 Alternating current 71 Ampere 30 Anticyclone 47 Area 142 Areal axis 198 Argand 4 Ausdehnungslehre 3, 9 Average velocity 57 Axial vector 30 Barycentric calculus 8 Bigelow 50, 60 Biquaternions 3, 126 Biradials 94 Bivector 29 Bjerknes 48, 57, 59, 290 Cailler 2 Cardan 3 Center (singularity) 44 Center of isogons 48 Change of basis 54 Characteristic equation 125 Characteristic equation of dyadic 221 Chi of dyadic 235 Christoffel's conditions 266 Circuital derivative 167 Circular multiplication 9 Circulation 78, 129 Clifford 3, 90 Combebiac 3 Complex numbers 63 Congruences 51, 138 Conjugate 66 Conjugate function 5 Continuous group 195 Continuous plane media 87 Convergence 177 Coulomb 13 Couple 139 Crystals 109 Cubic dilatation 258 Curl 76,82, 184 Curl of field 77 Curvature 148, 152 Curves 148 Cycle 30, 37 Cyclone 47 Derivative dyad 242 Developables 150 Dickson 105 Differential of p . . . ' 145 Differential of q 155, 159 Differential of vector 55 Differentiator 248 Directional derivative 166 Discharge 130 Discontinuities 261 Dissipation (plane) 84 Dissipation, dispersion 180 Divergence 76, 82 Divergence of field 77 Dyadic 2, 11, 218 Dyadic field 246 Dyname . 2 Dyne 29 Electric current 30 Electric density current 30 Electric induction 32 Electric intensity 31, 139 Energy 14 Energy current 30 Energy-density 15, 131 Energy-density current 30 Energy flux 142 Equation of continuity 87 Equipollences 71 Equipotential 15 Erg 14 Euler 107 Exact differential 190 Exterior multiplication 9 Extremals 160 Eye of cyclone 47 311 :;il> VECTOR CALCULUS Farad 32, 73 Faux 37, 38 Faux-focus 44 Feuille 30 Feuillets 2 Field 13 Flow 142 Flux 29, 130, 142 Flux density 29 Focus 41 Force 29 Force density 28, 141 Force function 18 Franklin 90 Free vector 8, 25 Frenet-vSerret formulae 148 Functions of dyadic 238 Function of flow 88 Functions of quaternions. ... 121 Gas defined 87 Gauss 4/ Gauss (magnetic unit). . . .32, 130 Gaussian operator 108 General equation of dyadic . . 220 Geometric curl 76 Geometric divergence 76 Geometric loci 133 Geometric vector 1 Geometry of lines 2 Gibbs 2, 11, 215 Gilbert 32, 130, 143 Glissant 26 Gradient 16, 163 Gram 15 Grassmann 2, 3, 9 Green's Theorem 205 Groups 8 Guiot 138 Hamilton 2, 3, 4,65,95 Harmonics 84, 169 Heaviside 31 Henry (electric unit) 32, 73 Hertzian vectors 33 Hitchcock 49 Hodograph 27 Hypernumber 3, 94 Imaginary 65 Impedance 73 Inductance. 73 Inductivity 32 Integral of vector 56 Integrating factor 191 Integration by parts 198 Interior multiplication 10 Invariant line 219 Irrotational 88 Isobaric 15,288 Isogons 34 Isohydric 15 Isopycnic 15, 288 Isosteric 15, 288 Isothermal 15 Joly 138, 147 Joule 14 Joule-second 14 Kinematic compatibility .... 266 Kirchoff's laws .' 73 Koenig 198, 205 Laisant 71 Lamellae 15 Lamellar field 84, 181 Laplace's equation 214 Latent equation 220 Laws of quaternions 103 Leibniz 3 Level 15 Line (electric unit) 32, 130 Lineal multiplication 9 Linear associative algebra ... 3 Linear vector function 218 Line of centers 46 Line of convergence 47 Line of divergence 47 Line of fauces 46 Line of foci 46 Line of nodes 45 Lines as levels 80 Liquid defined 87 MacMahon 75 Magnetic current 31 Magnetic density current 31 Magnetic induction 32 Magnetic intensity 32, 139 Mass 15 Matrix unity 65 Maxwell 13 McAulay 3 Mobius 8 Modulus 66 Moment 138 Moment of momentum 139 INDEX 313 Momentum 28 Momentum density 28 Momentum of field 141 Monodromic 14 Monogenic 89 Moving electric field 140 Moving magnetic field 140 Multenions 3 Multiple 6 Mutation 108 Nabla as complex number. . . 82 Nabla in plane 80 Nabla in space 162 Neutral point 47 Node 37,38 Node of isogons 48 Non-degenerate equations . . . 225 Norm 66 Notations One vector 12 Scalar 127 Two vectors 136 Derivative of vectors 165 Divergence, vortex, deriva- tive dyads 179 Dyadics 248 Ohm (electric unit) 73 Orthogonal dyadic 241 Orthogonal transformation . . 55 Peirce, Benjamin 3 Peirce, B. O 85 Permittance 73 Permittivity 32 Phase angle 71 Plane fields.. 84 Poincare 36, 46 Polar vector 30 Polydromic 14 Potential ,. .. 15, 17 Progressive multiplication ... 10 Power 76 Poynting vector 141 Pressure 142 Product of quaternions 98 Product of several quater- nions 113 Product of vectors 101 Quantum 14 Quaternions 2, 3, 6, 7, 95 Radial 26 Radius vector 26 Ratio of vectors 62 Reactance 73 Real 65 Reflections \ 108 Refraction . . 112 Regressive multiplication. ... 10 Relative derivative 18 Right versor 96 Rotations 108 Rotatory deviation 175 Saint Venant's equations. . . . 260 Sandstrom 35, 49 Saussure 2 Scalar 13 Scalar invariants 220, 239 Scalar of q 96 Schouten 7 Science of extension 2 Self transverse 234 Servois 4 Shear 256 Similitude 242 Singularities of vector lines . . 244 Singular lines 45 Solenoidal field 84, 181 Solid angles 117 Solution of equations 123 Solution of differential equa- tions 195 Solution of linear equation. . . 229 Specific momentum 28 Spherical astronomy 110 Squirt 90 Steinmetz 68, 71 Stoke's theorem 200 Strain 253 Strength of source or sink ... 90 Stress 143,269 Study . . . 2 Sum of quaternions 96 Surfaces 151 Symmetric multiplication ... 9 Tensor 65 Tensor of q 96 Torque 140 Tortuosity 149 Trajectories 150 Transport 130,298 Transverse dyadic 231 Triplex 25 314 VECTOR CALCULUS Triquaternions 3 Trirectangular biradials 100 Unit tube 18 Vacuity 220 Vanishing invariants 240 Variable trihedral 172 Vector 1 Vector calculus 1, 25 Vector field 23, 26 Vector lines 33 Vector of q 96 Vector potential 33, 93, 181 Vector surfaces 34 Vector tubes 34 Velocity 27 Velocity potential 18 Versor 65 Versor of q 96 Virial 129 Volt 31, 130, 143 Vortex 92, 187, 187 Vorticity 247,304 Waterspouts 50 Watt 15 Weber 14 Wessel 4 Whirl 90 Zero roots of linear equations. 230 foist r Ot— C/ p^V A^y 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. ■ JSJ — ^o. JM181S8 - 1MIH r- — l^MM o^ p - DEC 9 1968 52 "HI ttulb'68-3PM LOAN DEPT. ^ ^ REC'DLD JUN ^ & & * 2 172-9PMK8 LD 21A-60m-2,'67 (H241slO)476B LD 21A-50to-11 '62 (D3279sl0)476B jMCWI General Library University of California Berkeley General Library University of California Berkelev